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b(analyzed)f(all)h(the)f(sequences)g(that)g(w) -36 b(e)485 b(do)g(men)-36 b(tion.)1600 30434 y(In)470 b(some)h(cases)f(w)-36 b(e)471 b(just)f(list)h(the)f(\257rst)f(few)i (terms)f(and)g(in)-36 b(vite)470 b(the)g(reader)g(to)g(in)-36 b(v)g(estigate)472 b(them)1600 32039 y(himself.)981 b(W)-108 b(e)568 b(giv)-36 b(e)568 b(the)f(iden)-36 b(ti\257cation)568 b(n)-36 b(um)g(b)36 b(ers)566 b(of)i(these)f(sequences)h(in)f([)p 0 1 0 0 TeXcolorcmyk(12)p (#cite.OEIS) [[441 429 453 441] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])h(|)f(the)g(en)-36 b(tries)1600 33644 y(there)433 b(will)i(b)36 b(e)433 b(up)36 b(dated)432 b(as)i(more)g(information)g (b)36 b(ecomes)434 b(a)-36 b(v)-72 b(ailable.)3551 35249 y(W)-108 b(e)556 b(ha)-36 b(v)g(e)556 b(also)h(in)-36 b(v)g(estigated)556 b(sequences)g(arising)h(when)f(\()p 0 1 0 0 TeXcolorcmyk(2)p (#equation.2) [[373 400 379 412] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))f(is)i(replaced)f(b)-36 b(y)555 b(the)h(follo)-36 b(wing)1600 36854 y(rule:)633 b Fm(s)p Fo(\(1\))415 b(=)g Fm(x;)221 b(s)p Fo(\()p Fm(n)p Fo(\))416 b(=)f Fm(s)p Fo(\()p Fm(n)313 b Fl(\241)h Fo(1\))g(+)f Fm(y)509 b Fo(if)462 b Fm(n)f Fo(is)g(already)h(in)e(the)h(sequence,)468 b Fm(s)p Fo(\()p Fm(n)p Fo(\))415 b(=)g Fm(s)p Fo(\()p Fm(n)313 b Fl(\241)h Fo(1\))g(+)g Fm(z)1600 38459 y Fo(otherwise,)434 b(for)g(sp)36 b(eci\257ed)433 b(v)-72 b(alues)434 b(of)h Fm(x;)221 b(y)48 b(;)221 b(z)59 b Fo(.)580 b(This)434 b(w)-36 b(ork)434 b(will)h(b)36 b(e)433 b(describ)36 b(ed)433 b(elsewhere)h([)p 0 1 0 0 TeXcolorcmyk(4)p (#cite.CSV03) [[493 371 499 383] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(].)1600 41923 y Fn(Notation.)1303 b Fo(\\Sequence")511 b(here)f(usually)i(means)f(an)g(in\257nite)f(sequence)h(of)h (nonnegativ)-36 b(e)511 b(n)-36 b(um-)1600 43528 y(b)36 b(ers.)1169 b(\\Monotonically)632 b(increasing")f(means)g(that)f(eac) -36 b(h)630 b(term)g(is)h(strictly)g(greater)g(than)e(the)1600 45133 y(previous)434 b(term.)578 b Fh(P)368 b Fo(=)h Fl(f)p Fo(1)p Fm(;)221 b Fo(2)p Fm(;)g 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b(b)36 b(e)484 b(1,)498 b(since)484 b(1)h(is)f(o)36 b(dd)484 b(and)g(1)h(w)-36 b(ould)484 b(b)36 b(e)484 b(in)h(the)e(sequence.)731 b(It)1600 61568 y(could)562 b(also)g(b)36 b(e)562 b(2,)594 b(since)562 b(then)f(1)h(w)-36 b(ould)561 b(not)h(b)36 b(e)561 b(in)h(the)f(sequence)h(\(b)36 b(ecause)561 b(the)g(terms)h(m) -36 b(ust)1600 63173 y(increase\))404 b(and)f(2)h(is)g(ev)-36 b(en.)569 b(But)403 b(w)-36 b(e)404 b(m)-36 b(ust)403 b(tak)-36 b(e)405 b(the)e Fp(smal)66 b(lest)404 b Fo(p)36 b(ossible)405 b(v)-72 b(alue,)410 b(so)404 b Fm(a)p Fo(\(1\))369 b(=)g(1.)568 b(No)-36 b(w)1600 64778 y Fm(a)p Fo(\(2\))455 b(cannot)g(b)36 b(e)456 b(2,)461 b(b)36 b(ecause)456 b(2)f(is)h(ev)-36 b(en.)644 b(Nor)456 b(can)g Fm(a)p Fo(\(2\))f(b)36 b(e)455 b(3,)462 b(for)456 b(then)e(2)i(w)-36 b(ould)456 b(not)f(b)36 b(e)455 b(in)g(the)1600 66383 y(sequence)547 b(but)f Fm(a)p Fo(\(2\))g(w)-36 b(ould)547 b(b)36 b(e)547 b(o)36 b(dd.)917 b(Ho)-36 b(w)g(ev)g(er,)576 b Fm(a)p Fo(\(2\))562 b(=)f(4)547 b(is)h(p)36 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(http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A079000) pdfm Black(\):)12671 4691 y Fm(n)369 b Fo(:)1107 b(1)h(2)f(3)g(4)h(5)f(6) 1757 b(7)h(8)g(9)1107 b(10)h(11)f(12)h Fl(\242)221 b(\242)g(\242)10976 6296 y Fm(a)p Fo(\()p Fm(n)p Fo(\))369 b(:)1107 b(1)h(4)f(6)g(7)h(8)f (9)g(11)h(13)f(15)h(16)g(17)f(18)h Fl(\242)221 b(\242)g(\242)3551 8913 y Fo(Once)398 b(w)-36 b(e)398 b(are)h(past)f Fm(a)p Fo(\(2\))g(there)f(are)i(no)f(further)f(complications,)407 b Fm(a)p Fo(\()p Fm(n)223 b Fl(\241)g Fo(1\))399 b(is)f(greater)h(than) e Fm(n)p Fo(,)1600 10518 y(and)433 b(w)-36 b(e)434 b Fp(c)-66 b(an)p Fo(,)433 b(and)g(therefore)h Fp(must)p Fo(,)g(tak)-36 b(e)20346 13152 y Fm(a)p Fo(\()p Fm(n)p Fo(\))369 b(=)f Fm(a)p Fo(\()p Fm(n)295 b Fl(\241)h Fo(1\))f(+)g Fm(\262)434 b(;)p Black 17305 w Fo(\(2\))p Black 1600 15785 a(where)f Fm(\262)h Fo(is)g(1)g(or)g(2)g(and)f(is)h(giv)-36 b(en)434 b(b)-36 b(y:)23890 18306 y Fm(a)p Fo(\()p Fm(n)295 b Fl(\241)g Fo(1\))434 b(ev)-36 b(en)1107 b Fm(a)p Fo(\()p Fm(n)295 b Fl(\241)g Fo(1\))434 b(o)36 b(dd)13942 20711 y Fm(n)434 b Fo(in)g(sequence)5791 b(1)7966 b(2)12786 22317 y Fm(n)434 b Fo(not)f(in)h(sequence)4635 b(2)7966 b(1)1600 24860 y(The)434 b(gap)f(b)36 b(et)-36 b(w)g(een)433 b(successiv)-36 b(e)435 b(terms)e(for)h Fm(n)369 b Fl(\270)g Fo(3)434 b(is)g(either)f(1)h(or)g(2.)3551 26465 y(The)448 b(analogy)j(with)d(Aronson's)h(sequence)f(is)h(clear.)624 b(Just)448 b(as)h(Aronson's)f(sen)-36 b(tence)448 b(indicates)1600 28070 y(exactly)435 b(whic)-36 b(h)433 b(of)i(its)e(terms)h(are)f(t's,) h Fl(f)p Fm(a)p Fo(\()p Fm(n)p Fo(\))p Fl(g)g Fo(indicates)g(exactly)h (whic)-36 b(h)433 b(of)h(its)g(terms)f(are)h(o)36 b(dd.)3551 29675 y(W)-108 b(e)434 b(pro)36 b(ceed)433 b(to)g(analyze)i(the)e(b)36 b(eha)-36 b(vior)434 b(of)h(this)e(sequence.)3551 31280 y(First,)393 b(all)384 b(o)36 b(dd)383 b(n)-36 b(um)g(b)36 b(ers)382 b Fl(\270)369 b Fo(7)383 b(o)36 b(ccur.)562 b(F)-108 b(or)383 b(supp)36 b(ose)382 b(2)p Fm(t)192 b Fo(+)g(1)384 b(is)f(missing.)562 b(Therefore)384 b Fm(a)p Fo(\()p Fm(i)p Fo(\))368 b(=)h(2)p Fm(t)p Fo(,)1600 32885 y Fm(a)p Fo(\()p Fm(i)213 b Fo(+)g(1\))368 b(=)h(2)p Fm(t)213 b Fo(+)g(2)393 b(for)h(some)g Fm(i)368 b Fl(\270)i Fo(3.)565 b(F)-108 b(rom)393 b(the)f(de\257nition,)401 b(this)393 b(means)h Fm(i)f Fo(and)g Fm(i)213 b Fo(+)g(1)393 b(are)g(missing,)1600 34490 y(implying)434 b(a)g(gap)g(of)g(at)g(least) g(3,)g(a)g(con)-36 b(tradiction.)3551 36095 y(T)-108 b(able)p 0 1 0 0 TeXcolorcmyk 434 w(I)p (#table.1) [[136 392 140 404] [1 1 1 [3 3]] [0 0 1]] pdfm Black 434 w(sho)-36 b(ws)433 b(the)g(\257rst)g(72)h(terms,)g(with)g (the)f(ev)-36 b(en)433 b(n)-36 b(um)g(b)36 b(ers)432 b(underlined.)3551 37701 y(Examining)395 b(the)e(table,)402 b(w)-36 b(e)395 b(see)f(that)f(there)g(are)i(three)e(consecutiv)-36 b(e)394 b(n)-36 b(um)g(b)36 b(ers,)401 b(6,)h(7,)h(8,)f(whic)-36 b(h)1600 39306 y(are)449 b(necessarily)h(follo)-36 b(w)g(ed)451 b(b)-36 b(y)449 b(three)f(consecutiv)-36 b(e)450 b(o)36 b(dd)449 b(n)-36 b(um)g(b)36 b(ers,)451 b Fm(a)p Fo(\(6\))395 b(=)g(9,)454 b Fm(a)p Fo(\(7\))395 b(=)g(11,)454 b Fm(a)p Fo(\(8\))395 b(=)1600 40911 y(13.)992 b(Th)-36 b(us)571 b(9)h(is)f(presen)-36 b(t,)605 b(10)572 b(is)g(missing,)606 b(11)572 b(is)g(presen)-36 b(t,)604 b(12)572 b(is)g(missing,)606 b(and)571 b(13)h(is)g(presen)-36 b(t.)1600 42516 y(Therefore)441 b(the)e(sequence)h(con)-36 b(tin)g(ues)439 b(with)h Fm(a)p Fo(\(9\))380 b(=)f(15)441 b(\(o)36 b(dd\),)441 b Fm(a)p Fo(\(10\))380 b(=)g(16)440 b(\(ev)-36 b(en\),)442 b Fm(:)221 b(:)g(:)q Fo(,)442 b Fm(a)p Fo(\(13\))380 b(=)1600 44121 y(19)488 b(\(o)36 b(dd\),)501 b Fm(a)p Fo(\(14\))461 b(=)g(20)488 b(\(ev)-36 b(en\).)741 b(This)487 b(b)36 b(eha)-36 b(vior)489 b(is)f(rep)36 b(eated)487 b(for)h(ev)-36 b(er.)741 b(A)488 b(run)e(of)j(consecutiv)-36 b(e)1600 45726 y(n)g(um)g(b)36 b(ers)310 b(is)i(immediately)h(follo)-36 b(w)g(ed)314 b(b)-36 b(y)311 b(a)i(run)d(of)j(the)e(same)h(length)g(of) h(consecutiv)-36 b(e)312 b(o)36 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b(=)f(12)388 b Fl(\242)f Fo(2)27231 51664 y Fi(k)28187 52146 y Fl(\241)g Fo(3)g(+)f Fm(j)75 b Fo(;)636 b(the)568 b(second)f(half,)603 b(where)568 b Fm(j)673 b Fl(\270)598 b Fo(0,)1600 53751 y(consists)453 b(of)h(consecutiv)-36 b(e)453 b(o)36 b(dd)453 b(n)-36 b(um)g(b)36 b(ers)451 b(giv)-36 b(en)454 b(b)-36 b(y)453 b Fm(a)p Fo(\()p Fm(n)p Fo(\))402 b(=)g(12)309 b Fl(\242)f Fo(2)35117 53269 y Fi(k)35995 53751 y Fl(\241)g Fo(3)h(+)f(2)p Fm(j)75 b Fo(.)637 b(W)-108 b(e)453 b(can)g(com)-36 b(bine)1600 55357 y(these)433 b(form)-36 b(ulae,)435 b(obtaining)e(an)h(explicit)g (description)f(for)h(the)f(sequence:)20533 57990 y Fm(a)p Fo(\(1\))369 b(=)f(1)p Fm(;)1523 b(a)p Fo(\(2\))369 b(=)f(4)p Fm(;)1600 60624 y Fo(and)433 b(subsequen)-36 b(t)432 b(terms)i(are)f(giv)-36 b(en)434 b(b)-36 b(y)14643 63776 y Fm(a)p Fo(\(9)295 b Fl(\242)g Fo(2)18091 63228 y Fi(k)18955 63776 y Fl(\241)h Fo(3)f(+)g Fm(j)75 b Fo(\))369 b(=)f(12)296 b Fl(\242)f Fo(2)28314 63228 y Fi(k)29179 63776 y Fl(\241)g Fo(3)h(+)32892 62877 y(3)p 32892 63471 651 54 v 32892 64687 a(2)33675 63776 y Fm(j)370 b Fo(+)36023 62877 y(1)p 36023 63471 V 36023 64687 a(2)36806 63776 y Fl(j)p Fm(j)75 b Fl(j)p Black 11381 w Fo(\(3\))p Black 1600 66924 a(for)434 b Fm(k)414 b Fl(\270)369 b Fo(0,)434 b Fl(\241)p Fo(3)296 b Fl(\242)f Fo(2)10817 66442 y Fi(k)11755 66924 y Fl(\267)370 b Fm(j)443 b(<)369 b Fo(3)296 b Fl(\242)f Fo(2)17780 66442 y Fi(k)18349 66924 y Fo(.)3551 68529 y(The)421 b(structure)f(of)j(this)e(sequence)g(is)h(further)f(rev)-36 b(ealed)422 b(b)-36 b(y)421 b(examining)h(the)f(sequence)h(of)g (\257rst)1600 70134 y(di\256erences,)433 b(\242)p Fm(a)p Fo(\()p Fm(n)p Fo(\))369 b(=)f Fm(a)p Fo(\()p Fm(n)296 b Fo(+)e(1\))i Fl(\241)f Fm(a)p Fo(\()p Fm(n)p Fo(\),)434 b Fm(n)369 b Fl(\270)g Fo(1,)434 b(whic)-36 b(h)434 b(is)14376 72767 y(3)p Fm(;)221 b Fo(2)p Fm(;)g Fo(1)p Fm(;)g Fo(1)p Fm(;)g Fo(1)p Fm(;)g Fo(2)p Fm(;)g Fo(2)p Fm(;)g Fo(2)p Fm(;)g Fo(1)24882 72219 y Fk(6)25417 72767 y Fm(;)g Fo(2)26649 72219 y Fk(6)27176 72767 y Fm(;)g Fo(1)28408 72219 y Fk(12)29405 72767 y Fm(;)g Fo(2)30637 72219 y Fk(12)31634 72767 y Fm(;)g Fo(1)32866 72219 y Fk(24)33863 72767 y Fm(;)g Fo(2)35095 72219 y Fk(24)36093 72767 y Fm(;)g(:)g(:)g(:)p Black 11338 w Fo(\(4\))p Black Black 26075 76088 a(3)p Black eop %%Page: 4 4 4 3 bop Black 0 TeXcolorgray Black Black Black 14456 2002 a Fm(n)369 b Fo(:)1758 b(1)f(2)p 21309 2483 45 1606 v 1758 w(3)h(4)f(5)h(6)f(7)h(8)p 35755 2483 V 1758 w(9)1107 b(10)12761 3607 y Fm(a)p Fo(\()p Fm(n)p Fo(\))369 b(:)1758 b(1)f(4)p 20127 3819 651 54 v 21309 4088 45 1606 v 1758 w(6)p 22535 3819 651 54 v 1758 w(7)g(8)p 27350 3819 V 1758 w(9)1107 b(11)h(13)p 35755 4088 45 1606 v 1107 w(15)g(16)p 38738 3819 1301 54 v 14456 6817 a Fm(n)369 b Fo(:)1107 b(11)h(12)g(13)f(14)h(15)f(16)h(17)g(18)f(19)h(20)p 40570 7298 45 1606 v 12761 8422 a Fm(a)p Fo(\()p Fm(n)p Fo(\))369 b(:)1107 b(17)h(18)p 19477 8635 1301 54 v 1108 w(19)f(20)p 24292 8635 V 1108 w(21)g(23)h(25)g(27)f(29)h(31)p 40570 8904 45 1606 v 14456 11632 a Fm(n)369 b Fo(:)1107 b(21)h(22)g(23)f(24)h (25)f(26)h(27)g(28)f(29)h(30)12761 13237 y Fm(a)p Fo(\()p Fm(n)p Fo(\))369 b(:)1107 b(33)h(34)p 19477 13450 1301 54 v 1108 w(35)f(36)p 24292 13450 V 1108 w(37)g(38)p 29107 13450 V 1108 w(39)h(40)p 33923 13450 V 1107 w(41)g(42)p 38738 13450 V 14456 16447 a Fm(n)369 b Fo(:)1107 b(31)h(32)g(33)f(34)h (35)f(36)h(37)g(38)f(39)h(40)12761 18053 y Fm(a)p Fo(\()p Fm(n)p Fo(\))369 b(:)1107 b(43)h(44)p 19477 18265 V 1108 w(45)f(47)h(49)f(51)h(53)g(55)f(57)h(59)14456 21263 y Fm(n)369 b Fo(:)1107 b(41)h(42)g(43)f(44)p 26124 21744 45 1606 v 1108 w(45)g(46)h(47)g(48)f(49)h(50)12761 22868 y Fm(a)p Fo(\()p Fm(n)p Fo(\))369 b(:)1107 b(61)h(63)g(65)f(67)p 26124 23349 V 1108 w(69)g(70)p 29107 23080 1301 54 v 1108 w(71)h(72)p 33923 23080 V 1107 w(73)g(74)p 38738 23080 V 14456 26078 a Fm(n)369 b Fo(:)1107 b(51)h(52)g(53)f(54)h(55)f (56)h(57)g(58)f(59)h(60)12761 27683 y Fm(a)p Fo(\()p Fm(n)p Fo(\))369 b(:)1107 b(75)h(76)p 19477 27896 V 1108 w(77)f(78)p 24292 27896 V 1108 w(79)g(80)p 29107 27896 V 1108 w(81)h(82)p 33923 27896 V 1107 w(83)g(84)p 38738 27896 V 14456 30893 a Fm(n)369 b Fo(:)1107 b(61)h(62)g(63)f(64)h(65)f (66)h(67)g(68)f(69)h(70)12761 32498 y Fm(a)p Fo(\()p Fm(n)p Fo(\))369 b(:)1107 b(85)h(86)p 19477 32711 V 1108 w(87)f(88)p 24292 32711 V 1108 w(89)g(90)p 29107 32711 V 1108 w(91)h(92)p 33923 32711 V 1107 w(93)g(95)14456 35709 y Fm(n)369 b Fo(:)1107 b(71)h(72)g Fl(\242)221 b(\242)g(\242)12761 37314 y Fm(a)p Fo(\()p Fm(n)p Fo(\))369 b(:)1107 b(97)h(99)g Fl(\242)221 b(\242)g(\242)1600 40729 y Fo(T)-108 b(able)490 b(I:)p 0 TeXcolorgray Black 491 w(The)g(\257rst)g(72)h(terms)e(of)i(the)f(sequence)g(\\)p Fm(n)h Fo(is)f(in)h(the)e(sequence)h(if)h(and)f(only)h(if)g Fm(a)p Fo(\()p Fm(n)p Fo(\))f(is)1600 42334 y(o)36 b(dd.")p Black 1600 45692 a(\()p 0 1 0 0 TeXcolorcmyk(A79948)p [[91 306 129 318] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A079948) pdfm Black(\),)451 b(where)445 b(w)-36 b(e)447 b(ha)-36 b(v)g(e)446 b(written)g(1)21572 45210 y Fi(m)22905 45692 y Fo(to)g(indicate)g(a)h (string)f(of)g Fm(m)g Fo(1's,)k(etc.)615 b(The)446 b(oscillations)1600 47297 y(double)433 b(in)g(length)h(at)f(eac)-36 b(h)434 b(step.)3551 48902 y(Segmen)-36 b(t)399 b(0)h(b)36 b(egins)399 b(with)h(an)f(ev)-36 b(en)400 b(n)-36 b(um)g(b)36 b(er,)405 b(6,)i(but)398 b(all)i(other)g(segmen)-36 b(ts)399 b(b)36 b(egin)399 b(with)h(an)f(o)36 b(dd)1600 50507 y(n)-36 b(um)g(b)36 b(er,)515 b(9)340 b Fl(\242)g Fo(2)9126 50025 y Fi(k)10035 50507 y Fl(\241)g Fo(3.)776 b(All)500 b(o)36 b(dd)499 b(n)-36 b(um)g(b)36 b(ers)498 b(o)36 b(ccur)500 b(in)f(the)g(sequence)g(except)h(3)f(and)g(5.)776 b(The)500 b(ev)-36 b(en)1600 52112 y(n)g(um)g(b)36 b(ers)432 b(that)h(o)36 b(ccur)434 b(are)f(4,)i(6,)f(8)g(and)f(all)h(n)-36 b(um)g(b)36 b(ers)432 b(2)p Fm(m)i Fo(with)15552 54723 y(9)295 b Fl(\242)h Fo(2)17812 54174 y Fi(k)24 b Fj(\241)p Fk(1)19878 54723 y Fl(\241)296 b Fo(1)369 b Fl(\267)g Fm(m)g Fl(\267)g Fo(6)295 b Fl(\242)h Fo(2)28797 54174 y Fi(k)29661 54723 y Fl(\241)f Fo(2)p Fm(;)1523 b(k)414 b Fl(\270)369 b Fo(1)221 b Fm(:)3551 57334 y Fo(The)600 b(sequence)f(of)h (di\256erences,)641 b(\()p 0 1 0 0 TeXcolorcmyk(4)p (#equation.4) [[261 201 267 213] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\),)h(can)599 b(b)36 b(e)600 b(constructed)e(from)i(the)f(w)-36 b(ords)600 b(in)f(a)h(certain)1600 58939 y(formal)532 b(language)g(\(cf.)871 b([)p 0 1 0 0 TeXcolorcmyk(11)p (#cite.Lot83) [[201 187 213 199] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(]\).)g(Let)531 b(the)f(alphab)36 b(et)531 b(b)36 b(e)531 b Fl(A)g Fo(=)j Fl(f)p Fo(1)p Fm(;)221 b Fo(2)p Fm(;)g Fo(3)p Fl(g)p Fo(,)559 b(and)530 b(let)h Fl(A)44048 58457 y Fj(\244)45105 58939 y Fo(denote)f(the)1600 60544 y(set)487 b(of)i(strings)e(of)i(elemen)-36 b(ts)487 b(from)h Fl(A)p Fo(.)740 b(W)-108 b(e)487 b(de\257ne)g(a)h(mapping)f Fm(\265)524 b Fo(from)488 b Fl(A)39634 60062 y Fj(\244)40647 60544 y Fo(to)g Fl(A)43352 60062 y Fj(\244)44365 60544 y Fo(b)-36 b(y)487 b(the)g(rules)1600 62149 y Fm(\265)36 b Fo(\(1\))369 b(=)g(2)p Fm(;)221 b(\265)36 b Fo(\(2\))370 b(=)f(1)p Fm(;)221 b Fo(1)g(.)580 b(Then)433 b(\()p 0 1 0 0 TeXcolorcmyk(4)p (#equation.4) [[233 158 239 170] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))g(is)h(the)f(concatenation)20835 64760 y Fm(S)21635 64959 y Fj(\241)p Fk(1)22893 64760 y Fm(;)221 b(S)24275 64959 y Fk(0)24801 64760 y Fm(;)g(S)26183 64959 y Fk(1)26709 64760 y Fm(;)g(S)28091 64959 y Fk(2)28618 64760 y Fm(;)g(:)g(:)g(:)657 b(;)p Black 17795 w Fo(\(5\))p Black 1600 67371 a(where)11264 68976 y Fm(S)12064 69175 y Fj(\241)p Fk(1)13690 68976 y Fo(=)369 b Fl(f)p Fo(3)p Fm(;)221 b Fo(2)p Fl(g)p Fm(;)657 b(S)20099 69175 y Fk(0)20993 68976 y Fo(=)369 b Fl(f)p Fo(1)p Fm(;)221 b Fo(1)p Fm(;)g Fo(1)p Fl(g)p Fm(;)658 b(S)28635 69175 y Fi(k)24 b Fk(+1)30775 68976 y Fo(=)369 b Fm(\265)36 b Fo(\()p Fm(S)34104 69175 y Fi(k)34672 68976 y Fo(\))434 b(for)g Fm(k)414 b Fl(\270)369 b Fo(0)434 b Fm(:)p Black 8002 w Fo(\(6\))p Black 1600 71162 a(T)-108 b(o)383 b(pro)-36 b(v)g(e)383 b(this,)393 b(note)383 b(that)g(for)g Fm(n)370 b Fl(\270)f Fo(3,)394 b(a)383 b(di\256erence)f(of)i(2)f(only)h(o)36 b(ccurs)383 b(in)g Fl(f)p Fm(a)p Fo(\()p Fm(n)p Fo(\))p Fl(g)g Fo(b)36 b(et)-36 b(w)g(een)383 b(a)g(pair)1600 72767 y(of)399 b(o)36 b(dd)399 b(n)-36 b(um)g(b)36 b(ers.)565 b(Supp)36 b(ose)398 b Fm(a)p Fo(\()p Fm(i)p Fo(\))368 b(=)g(2)p Fm(j)300 b Fo(+)223 b(1)p Fm(;)e(a)p Fo(\()p Fm(i)k Fo(+)e(1\))370 b(=)e(2)p Fm(j)299 b Fo(+)224 b(3;)411 b(then)398 b Fm(a)p Fo(\(2)p Fm(j)299 b Fo(+)223 b(1\))369 b(=)g(2)p Fm(x)225 b Fo(+)e(1)399 b(\(sa)-36 b(y\),)p Black 26075 76088 a(4)p Black eop %%Page: 5 5 5 4 bop Black 0 TeXcolorgray Black Black 1600 2096 a Fm(a)p Fo(\(2)p Fm(j)299 b Fo(+)224 b(2\))369 b(=)g(2)p Fm(x)225 b Fo(+)f(2)p Fm(;)d(a)p Fo(\(2)p Fm(j)300 b Fo(+)224 b(3\))369 b(=)f(2)p Fm(x)225 b Fo(+)f(3,)407 b(pro)36 b(ducing)398 b(t)-36 b(w)g(o)399 b(di\256erences)f(of)i(1.)567 b(Similarly)-108 b(,)407 b(if)400 b(there)1600 3701 y(is)429 b(a)f(di\256erence)g(of)h(1,)g(sa)-36 b(y)429 b Fm(a)p Fo(\()p Fm(i)p Fo(\))369 b(=)f Fm(j)r(;)221 b(a)p Fo(\()p Fm(i)285 b Fo(+)f(1\))369 b(=)g Fm(j)359 b Fo(+)284 b(1,)430 b(then)d Fm(a)p Fo(\()p Fm(j)75 b Fo(\))368 b(=)h(2)p Fm(x)284 b Fo(+)g(1)p Fm(;)221 b(a)p Fo(\()p Fm(j)360 b Fo(+)284 b(1\))369 b(=)g(2)p Fm(x)284 b Fo(+)g(3,)1600 5306 y(a)434 b(di\256erence)f(of)h(2.)3551 6911 y(The)454 b(ratio)h Fm(n=a)p Fo(\()p Fm(n)p Fo(\),)460 b(whic)-36 b(h)454 b(is)h(the)f(fraction)h(of)g(p)36 b(ositiv)-36 b(e)455 b(in)-36 b(tegers)454 b(in)g(the)g(sequence)g(that)g(are)1600 8516 y(less)318 b(than)e(or)h(equal)h(to)g Fm(a)p Fo(\()p Fm(n)p Fo(\),)340 b(rises)318 b(from)f(close)h(to)g(2)p Fm(=)p Fo(3)g(at)f(the)g(b)36 b(eginning)317 b(of)h(segmen)-36 b(t)317 b Fm(k)362 b Fo(\(assuming)1600 10121 y Fm(k)518 b Fo(is)474 b(large\),)484 b(reac)-36 b(hes)473 b(a)g(maxim)-36 b(um)473 b(3)p Fm(=)p Fo(4)i(at)e(the)g(midp)36 b(oin)-36 b(t)472 b(of)i(the)f(segmen)-36 b(t,)483 b(then)472 b(falls)i(bac)-36 b(k)474 b(to)1600 11726 y(2)p Fm(=)p Fo(3)456 b(at)f(the)f(end)g(of)i (the)e(segmen)-36 b(t.)642 b(It)455 b(is)g(not)f(di\261cult)h(to)g(sho) -36 b(w)455 b(that)f(if)i Fm(n)f Fo(is)g(c)-36 b(hosen)454 b(at)h(random)1600 13331 y(in)429 b(the)g Fm(k)6070 12849 y Fk(th)7444 13331 y Fo(segmen)-36 b(t)429 b(then)f(the)h(a)-36 b(v)g(erage)430 b(v)-72 b(alue)430 b(of)g(the)f(fraction)h(of)g(n)-36 b(um)g(b)36 b(ers)428 b(in)i(the)e(sequence)i(at)1600 14936 y(that)j(p)36 b(oin)-36 b(t)433 b(approac)-36 b(hes)19405 16162 y(3)p 19405 16755 651 54 v 19405 17971 a(4)20484 17060 y Fl(\241)21945 16162 y Fo(1)p 21945 16755 V 21945 17971 a(4)22949 17060 y(log)24984 16162 y(32)p 24984 16755 1301 54 v 24984 17971 a(27)26786 17060 y(=)369 b(0)p Fm(:)p Fo(7075)221 b Fm(:)g(:)g(:)1600 19724 y Fo(for)434 b(large)h Fm(k)45 b Fo(.)3551 21329 y(The)434 b(sequence)f(has)h(an)f(alternativ)-36 b(e)434 b(c)-36 b(haracterization)434 b(in)g(terms)f(of)h(its)g(\\square.")3551 22934 y(The)461 b Fp(squar)-66 b(e)460 b Fo(of)h(a)g(sequence)g Fn(s)415 b Fo(=)g Fl(f)p Fm(s)p Fo(\()p Fm(n)p Fo(\))g(:)g Fm(n)g Fl(\270)g Fm(n)28421 23133 y Fk(0)28948 22934 y Fl(g)460 b Fo(is)h(giv)-36 b(en)461 b(b)-36 b(y)461 b Fn(s)37253 22452 y Fk(\(2\))38926 22934 y Fo(=)414 b Fl(f)p Fm(s)p Fo(\()p Fm(s)p Fo(\()p Fm(n)p Fo(\)\))h(:)g Fm(n)g Fl(\270)g Fm(n)49648 23133 y Fk(0)50174 22934 y Fl(g)p Fo(.)1600 24539 y(If)434 b Fn(s)g Fo(is)g(monotonically)h (increasing)f(so)g(is)g Fn(s)23220 24057 y Fk(\(2\))24478 24539 y Fo(.)p Black 1600 27583 a Fn(Lemma)499 b(1.)p Black 651 w Fp(L)-66 b(et)552 b Fn(s)h Fp(b)-66 b(e)552 b(monotonic)-66 b(al)66 b(ly)552 b(incr)-66 b(e)g(asing.)859 b(Then)552 b Fm(n)i Fo(\()p Fl(\270)532 b Fm(n)36895 27782 y Fk(0)37421 27583 y Fo(\))553 b Fp(is)g(in)f(the)h(se)-66 b(quenc)g(e)551 b Fn(s)j Fp(if)1600 29189 y(and)465 b(only)g(if)f Fm(s)p Fo(\()p Fm(n)p Fo(\))g Fp(is)h(in)f(the)h(se)-66 b(quenc)g(e)463 b Fn(s)22091 28706 y Fk(\(2\))23349 29189 y Fp(.)1600 32652 y Fn(Pro)42 b(of.)1301 b Fo(If)454 b Fm(n)h Fo(is)f(in)g(the)g(sequence,)460 b Fm(n)404 b Fo(=)g Fm(s)p Fo(\()p Fm(i)p Fo(\))453 b(for)h(some)h Fm(i)404 b Fl(\270)g Fm(n)33862 32851 y Fk(0)34388 32652 y Fo(,)460 b(and)453 b Fm(s)p Fo(\()p Fm(n)p Fo(\))404 b(=)g Fm(s)p Fo(\()p Fm(s)p Fo(\()p Fm(i)p Fo(\)\))452 b(is)j(in)f Fn(s)49581 32170 y Fk(\(2\))50839 32652 y Fo(.)1600 34257 y(Con)-36 b(v)g(ersely)-108 b(,)494 b(if)482 b Fm(s)p Fo(\()p Fm(n)p Fo(\))449 b Fl(2)g Fn(s)14509 33775 y Fk(\(2\))15767 34257 y Fo(,)493 b Fm(s)p Fo(\()p Fm(n)p Fo(\))450 b(=)f Fm(s)p Fo(\()p Fm(s)p Fo(\()p Fm(i)p Fo(\)\))479 b(for)j(some)f Fm(i)449 b Fl(\270)h Fm(n)33599 34456 y Fk(0)34125 34257 y Fo(,)493 b(and)481 b(since)g Fn(s)g Fo(is)g(monotonically)1600 35863 y(increasing,)434 b Fm(n)370 b Fo(=)e Fm(s)p Fo(\()p Fm(i)p Fo(\).)p 14294 35806 458 458 v 3551 37468 a(F)-108 b(or)289 b(our)f(sequence)h Fn(a)370 b Fo(=)e Fl(f)p Fm(a)p Fo(\()p Fm(n)p Fo(\))p Fl(g)p Fo(,)319 b(examination)290 b(of)g(T)-108 b(able)p 0 1 0 0 TeXcolorcmyk 289 w(I)p (#table.1) [[361 380 365 392] [1 1 1 [3 3]] [0 0 1]] pdfm Black 289 w(sho)-36 b(ws)289 b(that)g Fn(a)39885 36986 y Fk(\(2\))41512 37468 y Fo(=)369 b Fl(f)p Fo(1)p Fm(;)221 b Fo(5)p Fm(;)g Fo(7)p Fm(;)g Fo(9)p Fm(;)g Fo(11)p Fm(;)g(:)g(:)g(:)7 b Fl(g)369 b Fo(=)1600 39073 y Fl(f)p Fo(1)p Fl(g)319 b([)e Fo(2)p Fh(P)h Fo(+)g(3.)679 b(This)467 b(can)g(b)36 b(e)467 b(used)g(to)g(c)-36 b(haracterize)467 b Fn(a)p Fo(.)679 b(More)468 b(precisely)-108 b(,)476 b(the)466 b(sequence)h(can)g(b)36 b(e)1600 40678 y(de\257ned)559 b(b)-36 b(y:)833 b Fm(a)p Fo(\(1\))585 b(=)g(1,)593 b Fm(a)p Fo(\(2\))585 b(=)g(4,)593 b Fm(a)p Fo(\(3\))585 b(=)g(6)561 b(and,)592 b(for)562 b Fm(n)585 b Fl(\270)h Fo(4,)593 b Fm(a)p Fo(\()p Fm(n)p Fo(\))560 b(is)h(the)g(smallest)g(p) 36 b(osi-)1600 42283 y(tiv)-36 b(e)525 b(in)-36 b(teger)525 b(whic)-36 b(h)525 b(is)g(consisten)-36 b(t)525 b(with)g(the)f (sequence)h(b)36 b(eing)525 b(monotonically)i(increasing)e(and)1600 43888 y(satisfying)435 b Fm(a)p Fo(\()p Fm(a)p Fo(\()p Fm(n)p Fo(\)\))368 b(=)h(2)p Fm(n)296 b Fo(+)f(3)434 b(for)g Fm(n)369 b Fl(\270)g Fo(2.)3551 45493 y(This)502 b(is)h(easily)g(c)-36 b(hec)g(k)g(ed.)784 b(Once)501 b(the)h(\257rst)f(three)g(terms)h(are)g(sp)36 b(eci\257ed,)519 b(the)502 b(rule)g Fm(a)p Fo(\()p Fm(a)p Fo(\()p Fm(n)p Fo(\)\))484 b(=)1600 47098 y(2)p Fm(n)296 b Fo(+)f(3)434 b(determines)e(the)h(remaining)h(terms)f(uniquely)-108 b(.)3551 48703 y(In)633 b(fact)h(that)f(rule)g(also)i(forces)f Fm(a)p Fo(\(2\))f(to)g(b)36 b(e)634 b(4,)683 b(but)633 b(it)g(do)36 b(es)634 b(not)f(determine)g Fm(a)p Fo(\(3\),)683 b(since)1600 50308 y(there)578 b(is)i(an)e(earlier)i(sequence)f Fl(f)p Fm(a)19477 49826 y Fj(0)19788 50308 y Fo(\()p Fm(n)p Fo(\))p Fl(g)g Fo(\(in)f(the)h(lexicographic)h(sense\))f (satisfying)h Fm(a)45971 49826 y Fj(0)46282 50308 y Fo(\(1\))616 b(=)g(1,)1600 51914 y Fm(a)2283 51431 y Fj(0)2593 51914 y Fo(\()p Fm(a)3782 51431 y Fj(0)4092 51914 y Fo(\()p Fm(n)p Fo(\)\))369 b(=)g(2)p Fm(n)296 b Fo(+)f(3)434 b(for)g Fm(n)369 b Fl(\270)g Fo(2,)434 b(namely)13788 54847 y(1)p Fm(;)221 b Fo(4)p Fm(;)g Fo(5)p Fm(;)g Fo(7)p Fm(;)g Fo(9)p Fm(;)g Fo(10)p Fm(;)g Fo(11)p Fm(;)g Fo(12)p Fm(;)g Fo(13)p Fm(;)g Fo(15)p Fm(;)g Fo(17)p Fm(;)g Fo(19)p Fm(;)g Fo(21)p Fm(;)g Fo(22)p Fm(;)g(:)g(:)g(:)240 b(;)1600 57780 y Fo(\()p 0 1 0 0 TeXcolorcmyk(A80596)p [[91 197 129 209] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A080596) pdfm Black(\),)435 b(and)e(giv)-36 b(en)434 b(b)-36 b(y)434 b Fm(a)16048 57298 y Fj(0)16358 57780 y Fo(\(1\))369 b(=)g(1,)14813 61306 y Fm(a)15496 60758 y Fj(0)15806 61306 y Fo(\(6)295 b Fl(\242)h Fo(2)18572 60758 y Fi(k)19436 61306 y Fl(\241)f Fo(3)h(+)f Fm(j)75 b Fo(\))368 b(=)h(8)295 b Fl(\242)h Fo(2)28145 60758 y Fi(k)29009 61306 y Fl(\241)f Fo(3)h(+)32722 60408 y(3)p 32722 61001 651 54 v 32722 62217 a(2)33506 61306 y Fm(j)370 b Fo(+)35853 60408 y(1)p 35853 61001 V 35853 62217 a(2)36636 61306 y Fl(j)p Fm(j)75 b Fl(j)p Black 11551 w Fo(\(7\))p Black 1600 64753 a(for)434 b Fm(k)414 b Fl(\270)369 b Fo(0,)434 b Fl(\241)p Fo(2)9207 64271 y Fi(k)24 b Fk(+1)11348 64753 y Fl(\267)369 b Fm(j)444 b(<)369 b Fo(2)15763 64271 y Fi(k)24 b Fk(+1)17534 64753 y Fo(.)3551 66359 y(As)393 b(the)g(ab)36 b(o)-36 b(v)g(e)394 b(examples)g(sho)-36 b(w,)402 b(the)393 b(square)h(of)g(a)g(sequence)f (do)36 b(es)394 b(not)f(in)g(general)h(determine)1600 67964 y(the)561 b(sequence)h(uniquely)-108 b(.)964 b(A)562 b(b)36 b(etter)561 b(w)-36 b(a)g(y)563 b(to)f(do)g(this)g(is)g(pro)-36 b(vided)562 b(b)-36 b(y)561 b(the)h(\\in)-36 b(v)g(erse)562 b(Aronson)1600 69569 y(transform",)434 b(discussed)f(in)h(Section)p 0 1 0 0 TeXcolorcmyk 433 w(5)p (#section.5) [[255 91 261 103] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)p Black 26075 76088 a(5)p Black eop %%Page: 6 6 6 5 bop Black 0 TeXcolorgray Black Black 1600 2096 a Fq(3.)1614 b(First)716 b(generalizations)1600 5016 y Fo(The)373 b(prop)36 b(erties)373 b(of)h Fl(f)p Fm(a)p Fo(\()p Fm(n)p Fo(\))p Fl(g)g Fo(giv)-36 b(en)374 b(in)f(Section)p 0 1 0 0 TeXcolorcmyk 373 w(2)p (#section.2) [[298 672 304 684] [1 1 1 [3 3]] [0 0 1]] pdfm Black 374 w(suggest)h(man)-36 b(y)373 b(generalizations,)387 b(some)374 b(of)g(whic)-36 b(h)1600 6621 y(will)435 b(b)36 b(e)433 b(discussed)g(in)g(this)h(and)f(the)g(follo)-36 b(wing)436 b(sections.)1600 10074 y Fn(\(3.1\))650 b(Non-monotonic)i(v) -42 b(ersion.)1302 b Fo(If)565 b(w)-36 b(e)566 b(replace)f(\\)p Fm(a)p Fo(\()p Fm(n)p Fo(\))593 b Fm(>)f(a)p Fo(\()p Fm(n)384 b Fl(\241)h Fo(1\)")566 b(in)e(the)h(de\257nition)1600 11679 y(b)-36 b(y)572 b(\\)p Fm(a)p Fo(\()p Fm(n)p Fo(\))g(is)h(not)f (already)h(in)f(the)g(sequence",)607 b(w)-36 b(e)573 b(obtain)f(a)h(completely)g(di\256eren)-36 b(t)571 b(sequence,)1600 13284 y(suggested)379 b(b)-36 b(y)378 b(J.)i(C.)f(Lagarias)h([)p 0 1 0 0 TeXcolorcmyk(10)p (#cite.Lag03) [[232 597 244 609] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(]:)551 b Fm(b)p Fo(\()p Fm(n)p Fo(\),)390 b Fm(n)370 b Fl(\270)f Fo(1,)390 b(is)379 b(the)g(smallest)g(p)36 b(ositiv)-36 b(e)380 b(in)-36 b(teger)379 b(not)f(already)1600 14889 y(in)641 b(the)g(sequence)h(whic)-36 b(h)641 b(is)h(consisten)-36 b(t)641 b(with)h(the)f(condition)g(that)g(\\)p Fm(n)i Fo(is)e(a)h(mem)-36 b(b)36 b(er)641 b(of)h(the)1600 16494 y(sequence)434 b(if)g(and)f(only)h(if)g Fm(b)p Fo(\()p Fm(n)p Fo(\))g(is)g(o)36 b(dd.")578 b(This)434 b(sequence)g(\()p 0 1 0 0 TeXcolorcmyk(A79313)p [[355 569 393 581] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A079313) pdfm Black(\))g(b)36 b(egins:)14391 19310 y Fm(n)369 b Fo(:)1758 b(1)f(2)h(3)g(4)f(5)h(6)g(7)f(8)h(9)1107 b(10)12826 20916 y Fm(b)p Fo(\()p Fm(n)p Fo(\))369 b(:)1758 b(1)f(3)h(5)g(2)p 24878 21128 651 54 v 1757 w(7)g(8)p 29693 21128 V 1758 w(9)1107 b(11)g(13)h(12)p 38673 21128 1301 54 v 14391 24926 a Fm(n)369 b Fo(:)1107 b(11)h(12)g(13)f(14)h(15)g(16)f(17)h(18)f (19)h(20)12826 26531 y Fm(b)p Fo(\()p Fm(n)p Fo(\))369 b(:)1107 b(15)h(17)g(19)f(16)p 24227 26743 V 1108 w(21)h(23)f(25)h(20)p 33858 26743 V 1107 w(27)g(29)1600 29369 y(The)401 b(ev)-36 b(en)401 b(mem)-36 b(b)36 b(ers)400 b(are)h(underlined.)566 b(The)400 b(b)36 b(eha)-36 b(vior)402 b(is)f(simpler)g(than)f(that)g (of)i Fl(f)p Fm(a)p Fo(\()p Fm(n)p Fo(\))p Fl(g)p Fo(,)407 b(and)401 b(w)-36 b(e)1600 30975 y(lea)g(v)g(e)435 b(it)e(to)h(the)f (reader)g(to)h(sho)-36 b(w)434 b(that,)f(for)h Fm(n)370 b Fl(\270)f Fo(5,)434 b Fm(b)p Fo(\()p Fm(n)p Fo(\))f(is)h(giv)-36 b(en)435 b(b)-36 b(y)20209 33830 y Fm(b)p Fo(\(4)p Fm(t)295 b Fl(\241)g Fo(2\))1107 b(=)g(4)p Fm(t)221 b(;)20209 35767 y(b)p Fo(\(4)p Fm(t)295 b Fl(\241)g Fo(1\))1107 b(=)g(6)p Fm(t)295 b Fl(\241)h Fo(3)221 b Fm(;)22483 37704 y(b)p Fo(\(4)p Fm(t)p Fo(\))1106 b(=)h(6)p Fm(t)295 b Fl(\241)h Fo(1)221 b Fm(;)20230 39641 y(b)p Fo(\(4)p Fm(t)296 b Fo(+)e(1\))1107 b(=)g(6)p Fm(t)295 b Fo(+)g(1)221 b Fm(:)1600 42496 y Fo(All)415 b(o)36 b(dd)414 b(n)-36 b(um)g(b)36 b(ers)412 b(o)36 b(ccur.)572 b(The)414 b(only)h(ev)-36 b(en)414 b(n)-36 b(um)g(b)36 b(ers)413 b(are)h(2)h(and)f(4)p Fm(t)p Fo(,)k Fm(t)369 b Fl(\270)g Fo(2.)572 b(\(The)414 b(square)h Fn(b)48654 42014 y Fk(\(2\))50326 42496 y Fo(is)1600 44102 y(not)433 b(so)h(in)-36 b(teresting.\))1600 47554 y Fn(\(3.2\))698 b(\\Ev)-42 b(en")698 b(v)-42 b(ersion.)1302 b Fo(If)608 b(instead)f(w)-36 b(e)607 b(c)-36 b(hange)607 b(\\o)36 b(dd")608 b(in)f(the)f(de\257nition)h(of)g Fl(f)p Fm(a)p Fo(\()p Fm(n)p Fo(\))p Fl(g)h Fo(to)1600 49159 y(\\ev)-36 b(en",)607 b(w)-36 b(e)572 b(obtain)g(a)g(sequence)f Fn(c)h Fo(whic)-36 b(h)572 b(is)g(b)36 b(est)571 b(started)g(at)h Fm(n)604 b Fo(=)g(0:)855 b Fm(c)p Fo(\()p Fm(n)p Fo(\),)606 b Fm(n)e Fl(\270)h Fo(0,)h(is)572 b(the)1600 50764 y(smallest)434 b(nonnegativ)-36 b(e)434 b(in)-36 b(teger)434 b Fm(>)368 b(c)p Fo(\()p Fm(n)295 b Fl(\241)h Fo(1\))433 b(whic)-36 b(h)434 b(is)g(consisten)-36 b(t)433 b(with)g(the)g(condition)h(that) 9526 53619 y(\\)p Fm(n)g Fo(is)g(a)g(mem)-36 b(b)36 b(er)433 b(of)h(the)f(sequence)h(if)g(and)f(only)i(if)f Fm(c)p Fo(\()p Fm(n)p Fo(\))f(is)h(ev)-36 b(en.")p Black 6265 w(\(8\))p Black 1600 56474 a(This)334 b(is)p 0 1 0 0 TeXcolorcmyk 334 w(A79253)p [[123 209 161 221] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A079253) pdfm Black(:)530 b(0,)354 b(3,)g(5,)g(6,)g(7,)g(8,)h(10,)f(12,)h(14,)f(15,)g Fm(:)221 b(:)g(:)r Fo(.)545 b(It)334 b(is)g(easily)h(seen)e(that)g Fm(c)p Fo(\()p Fm(n)p Fo(\))369 b(=)f Fm(a)p Fo(\()p Fm(n)91 b Fo(+)g(1\))g Fl(\241)g Fo(1)1600 58079 y(for)434 b Fm(n)369 b Fl(\270)h Fo(0,)434 b(so)g(there)f(is)h(nothing)f(essen) -36 b(tially)434 b(new)g(here.)578 b(Also)434 b Fn(c)34401 57597 y Fk(\(2\))36028 58079 y Fo(=)369 b Fl(f)p Fo(0)p Fl(g)295 b([)g Fo(2)p Fh(P)g Fo(+)g(4.)1600 61532 y Fn(\(3.3\))466 b(The)f(\\lying")g(v)-42 b(ersion.)1302 b Fo(The)405 b(lying)h(v)-36 b(ersion)405 b(of)g(Aronson's)g(sequence)g(is)g(based)f (on)g(the)1600 63137 y(completely)492 b(false)g(sen)-36 b(tence)490 b(\\t)h(is)g(the)f(second,)505 b(third,)g(\257fth,)g (sixth,)h(sev)-36 b(en)g(th,)504 b Fm(:)221 b(:)g(:)493 b Fo(letter)e(of)g(this)1600 64742 y(sen)-36 b(tence.")557 b(The)372 b(sen)-36 b(tence)371 b(sp)36 b(eci\257es)372 b(exactly)h(those)e(letters)h(that)f(are)h(not)g(t's,)384 b(and)372 b(pro)36 b(duces)370 b(the)1600 66347 y(sequence)434 b(\()p 0 1 0 0 TeXcolorcmyk(A81023)p [[139 120 177 132] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A081023) pdfm Black(\))g(2,)h(3,)f(5,)g(6,)g(7,)g(8,)g(9,)h(10,)f(11,)h(12,)f Fm(:)221 b(:)g(:)r Fo(.)3551 67952 y(Just)608 b(as)g Fl(f)p Fm(a)p Fo(\()p Fm(n)p Fo(\))p Fl(g)g Fo(is)h(an)f(analogue)h(of) g(Aronson's)f(sequence,)652 b(w)-36 b(e)608 b(can)g(de\257ne)f(an)h (analogue)1600 69557 y Fl(f)p Fm(d)p Fo(\()p Fm(n)p Fo(\))578 b(:)g Fm(n)h Fl(\270)f Fo(1)p Fl(g)557 b Fo(of)g(this)f(sequence)h(b) -36 b(y)556 b(sa)-36 b(ying)557 b(that:)824 b Fm(d)p Fo(\()p Fm(n)p Fo(\))556 b(is)h(the)f(smallest)h(p)36 b(ositiv)-36 b(e)557 b(in)-36 b(teger)1600 71162 y Fm(>)451 b(d)p Fo(\()p Fm(n)328 b Fl(\241)g Fo(1\))482 b(suc)-36 b(h)481 b(that)g(the)h(condition)f(\\)p Fm(n)i Fo(is)f(in)g(the)f (sequence)h(if)h(and)e(only)h(if)h Fm(d)p Fo(\()p Fm(n)p Fo(\))f(is)g(o)36 b(dd")482 b(is)1600 72767 y(false.)753 b(Equiv)-72 b(alen)-36 b(tly)-108 b(,)507 b(the)490 b(condition)i (\\either)f Fm(n)h Fo(is)f(in)g(the)g(sequence)g(and)g Fm(d)p Fo(\()p Fm(n)p Fo(\))g(is)h(ev)-36 b(en)491 b(or)h Fm(n)g Fo(is)p Black 26075 76088 a(6)p Black eop %%Page: 7 7 7 6 bop Black 0 TeXcolorgray Black Black 1600 2096 a Fo(not)457 b(in)h(the)f(sequence)g(and)h Fm(d)p Fo(\()p Fm(n)p Fo(\))f(is)h(o)36 b(dd")458 b(should)f(b)36 b(e)457 b(true.)650 b(The)458 b(resulting)f(sequence)h(\()p 0 1 0 0 TeXcolorcmyk(A80653)p [[490 698 528 710] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A080653) pdfm Black(\))1600 3701 y(b)36 b(egins)471 b(2,)480 b(4,)h(5,)g(6,)f(8,)h (10,)g(11,)g(12,)g(13,)g(14,)g Fm(:)221 b(:)g(:)q Fo(.)690 b(W)-108 b(e)471 b(will)h(giv)-36 b(e)472 b(an)e(explicit)i(form)-36 b(ula)471 b(for)g Fm(d)p Fo(\()p Fm(n)p Fo(\))g(in)1600 5306 y(the)433 b(next)h(section.)3551 6911 y(A)470 b(related)h (sequence)f(is)h(also)g(of)g(in)-36 b(terest.)688 b(Let)470 b Fl(f)p Fm(d)29295 6429 y Fj(0)29606 6911 y Fo(\()p Fm(n)p Fo(\))p Fl(g)g Fo(b)36 b(e)470 b(de\257ned)f(b)-36 b(y)471 b Fm(d)41372 6429 y Fj(0)41682 6911 y Fo(\(1\))432 b(=)f(2,)480 b(and,)f(for)1600 8516 y Fm(n)369 b(>)g Fo(1,)389 b Fm(d)6202 8034 y Fj(0)6512 8516 y Fo(\()p Fm(n)p Fo(\))378 b(is)f(the)g(smallest)h(in)-36 b(teger)377 b(greater)h(than)e Fm(d)29306 8034 y Fj(0)29617 8516 y Fo(\()p Fm(n)180 b Fl(\241)g Fo(1\))377 b(suc)-36 b(h)377 b(that)g(the)f(condition)h(\\)p Fm(n)h Fo(and)1600 10121 y Fm(d)2276 9639 y Fj(0)2586 10121 y Fo(\()p Fm(d)3768 9639 y Fj(0)4079 10121 y Fo(\()p Fm(n)p Fo(\)\))451 b(ha)-36 b(v)g(e)453 b(opp)36 b(osite)452 b(parities")g(can)g(alw)-36 b(a)g(ys)454 b(b)36 b(e)451 b(satis\257ed.)634 b(One)451 b(can)h(sho)-36 b(w)452 b(that)g(this)f(is)h(the)1600 11726 y(sequence)17229 13331 y(2)p Fm(;)221 b Fo(4)p Fm(;)g Fo(5)p Fm(;)g Fo(7)p Fm(;)g Fo(8)p Fm(;)g Fo(9)p Fm(;)g Fo(11)p Fm(;)g Fo(12)p Fm(;)g Fo(13)p Fm(;)g Fo(14)p Fm(;)g Fo(16)p Fm(;)g(:)g(:)g(:)1600 15656 y Fo(\()p 0 1 0 0 TeXcolorcmyk(A14132)p [[91 576 129 588] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A014132) pdfm Black(\),)354 b(the)332 b(complemen)-36 b(t)332 b(of)i(the)d (triangular)i(n)-36 b(um)g(b)36 b(ers)331 b(\()p 0 1 0 0 TeXcolorcmyk(A217)p [[360 576 386 588] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000217) pdfm Black(\),)354 b(with)332 b Fm(d)39674 15174 y Fj(0)39985 15656 y Fo(\()p Fm(n)p Fo(\))701 b(=)g Fm(n)333 b Fo(+)f(\(nearest)1600 17261 y(in)-36 b(teger)433 b(to)7489 16162 y Fl(p)p 8596 16162 1427 54 v 1099 x Fo(2)p Fm(n)q Fo(\).)1600 21698 y Fq(4.)1614 b(The)716 b(\\mo)60 b(d)717 b Fg(m)p Fq(")h(v)-60 b(ersions)1600 24619 y Fo(Both)601 b Fl(f)p Fm(a)p Fo(\()p Fm(n)p Fo(\))p Fl(g)h Fo(and)f Fl(f)p Fm(c)p Fo(\()p Fm(n)p Fo(\))p Fl(g)g Fo(are)h(de\257ned)e(mo)36 b(dulo)602 b(2.)1082 b(Another)601 b(family)i(of)f(generalizations)h(is)1600 26224 y(based)549 b(on)f(replacing)i(2)f(b)-36 b(y)549 b(some)g(\257xed)g(in)-36 b(teger)548 b Fm(y)613 b Fl(\270)566 b Fo(2.)925 b(T)-108 b(o)549 b(this)g(end)f(w)-36 b(e)549 b(de\257ne)f(a)h(sequence)1600 27829 y Fl(f)p Fm(s)p Fo(\()p Fm(n)p Fo(\))369 b(:)g Fm(n)g Fl(\270)h Fm(n)9088 28028 y Fk(0)9614 27829 y Fl(g)421 b Fo(b)-36 b(y)421 b(sp)36 b(ecifying)423 b(a)e(starting)h(v)-72 b(alue)421 b Fm(s)p Fo(\()p Fm(n)29672 28028 y Fk(0)30198 27829 y Fo(\))369 b(=)f Fm(s)33066 28028 y Fk(0)33592 27829 y Fo(,)424 b(and)c(the)h(condition)g(that)g Fm(n)h Fo(is)f(in)1600 29434 y(the)433 b(sequence)h(if)g(and)f(only)h(if)g Fm(s)p Fo(\()p Fm(n)p Fo(\))369 b Fl(\264)g Fm(z)493 b Fo(\()74 b(mo)36 b(d)507 b Fm(y)48 b Fo(\),)434 b(where)f Fm(y)481 b Fo(and)434 b Fm(z)492 b Fo(are)434 b(giv)-36 b(en.)3551 31039 y(Although)476 b(w)-36 b(e)477 b(will)g(not)f(digress)h(to)f (consider)g(this)h(here,)486 b(it)477 b(is)g(also)g(of)g(in)-36 b(terest)476 b(to)g(see)h(what)1600 32644 y(happ)36 b(ens)496 b(when)h(\\if)i(and)e(only)h(if)93 b(")498 b(in)g(the)f(de\257nition)f (is)i(replaced)f(b)-36 b(y)498 b(either)f(\\if)93 b(")498 b(or)g(\\only)h(if.")1600 34249 y(\(W)-108 b(e)472 b(men)-36 b(tion)472 b(just)h(one)g(example.)696 b(The)472 b(ab)36 b(o)-36 b(v)g(e)474 b(sequence)e Fl(f)p Fm(d)p Fo(\()p Fm(n)p Fo(\))p Fl(g)p Fo(,)483 b(pre\257xed)472 b(b)-36 b(y)473 b Fm(d)p Fo(\(0\))435 b(=)g(0,)483 b(can)1600 35854 y(b)36 b(e)401 b(de\257ned)e(as)j(follo)-36 b(ws:)564 b Fm(d)p Fo(\()p Fm(n)p Fo(\))401 b(is)g(the)g(smallest)h(nonnegativ) -36 b(e)401 b(n)-36 b(um)g(b)36 b(er)400 b Fm(>)368 b(d)p Fo(\()p Fm(n)229 b Fl(\241)g Fo(1\))402 b(suc)-36 b(h)400 b(that)g(the)1600 37459 y(condition)433 b(\\)p Fm(n)i Fo(\()p Fm(n)369 b Fl(\270)g Fo(0\))434 b(is)g(in)f(the)g(sequence)h (only)g(if)g Fm(d)p Fo(\()p Fm(n)p Fo(\))g(is)g(ev)-36 b(en")433 b(is)h(satis\257ed.\))3551 39065 y(W)-108 b(e)361 b(sa)-36 b(w)362 b(in)f(the)f(previous)h(section)h(that)e Fl(f)p Fm(a)p Fo(\()p Fm(n)p Fo(\))p Fl(g)h Fo(can)g(also)h(b)36 b(e)361 b(c)-36 b(haracterized)361 b(b)-36 b(y)361 b(the)f(prop)36 b(ert)-36 b(y)1600 40670 y(that)386 b(its)h(square)g Fm(a)10864 40188 y Fk(\(2\))12121 40670 y Fo(\()p Fm(n)p Fo(\))369 b(=)g Fm(a)p Fo(\()p Fm(a)p Fo(\()p Fm(n)p Fo(\)\))386 b(is)g(equal)i(to)f(2)p Fm(n)199 b Fo(+)g(3)387 b(for)g Fm(n)370 b Fl(\270)f Fo(2)387 b(\(together)f(with)h(some)g (appro-)1600 42275 y(priate)495 b(initial)i(conditions\).)764 b(This)496 b(to)36 b(o)496 b(can)g(b)36 b(e)495 b(generalized)h(b)-36 b(y)496 b(sp)36 b(ecifying)497 b(that)e(the)g(sequence)1600 43880 y Fl(f)p Fm(s)p Fo(\()p Fm(n)p Fo(\))p Fl(g)397 b Fo(satisfy)i Fm(s)p Fo(\()p Fm(s)p Fo(\()p Fm(n)p Fo(\)\))368 b(=)g Fm(y)48 b(n)221 b Fo(+)g Fm(z)59 b Fo(,)406 b(for)397 b(giv)-36 b(en)398 b(v)-72 b(alues)398 b(of)g Fm(y)445 b Fo(and)397 b Fm(z)59 b Fo(.)566 b(The)398 b(t)-36 b(w)g(o)397 b(generalizations)i(are)1600 45485 y(related,)548 b(but)525 b(usually)h(lead)f(to)h(di\256eren)-36 b(t)524 b(sequences.)853 b(The)525 b Fm(s)p Fo(\()p Fm(s)p Fo(\()p Fm(n)p Fo(\)\))g(family)i(of) f(generalizations)1600 47090 y(will)369 b(connect)d(the)h(presen)-36 b(t)366 b(in)-36 b(v)g(estigation)369 b(with)e(sev)-36 b(eral)368 b(sequences)g(that)f(ha)-36 b(v)g(e)367 b(already)h(app)36 b(eared)1600 48695 y(in)403 b(the)g(literature.)568 b(There)404 b(are)f(to)36 b(o)404 b(man)-36 b(y)404 b(p)36 b(ossibilities)404 b(for)g(us)f(to)h(giv)-36 b(e)404 b(a)g(complete)g(catalogue)g(of)1600 50300 y(all)383 b(the)e(sequences)g(that)h(can)f(b)36 b(e)382 b(obtained)f(from)h(these)f(generalizations.)563 b(Instead)381 b(w)-36 b(e)382 b(will)h(giv)-36 b(e)383 b(a)1600 51905 y(few)433 b(k)-36 b(ey)433 b(examples)g(and)e(one)h (general)h(theorem.)577 b(Man)-36 b(y)432 b(other)g(examples)h(can)f(b) 36 b(e)432 b(found)f(in)h([)p 0 1 0 0 TeXcolorcmyk(12)p (#cite.OEIS) [[515 250 526 262] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(].)3551 53510 y(A)503 b(simple)h(\\mo)36 b(d)503 b(3")h(generalization)h(is:)718 b Fm(e)p Fo(\(1\))488 b(=)f(2,)522 b(and,)e(for)504 b Fm(n)489 b(>)e Fo(1,)521 b Fm(e)p Fo(\()p Fm(n)p Fo(\))504 b(is)f(the)g(smallest)1600 55115 y(in)-36 b(teger)433 b Fm(>)369 b(e)p Fo(\()p Fm(n)295 b Fl(\241)h Fo(1\))433 b(whic)-36 b(h)434 b(is)g(consisten)-36 b(t)433 b(with)g(the)g(condition)h(that)6614 58038 y(\\)p Fm(n)h Fo(is)e(a)h(mem)-36 b(b)36 b(er)433 b(of)i(the)e(sequence)g(if)h (and)f(only)i(if)f Fm(e)p Fo(\()p Fm(n)p Fo(\))f(is)h(a)g(m)-36 b(ultiple)433 b(of)i(3.")p Black 49538 58049 a(\(9\))p Black 1600 60982 a(This)f(turns)e(out)i(to)f(b)36 b(e)434 b(James)g(Propp's)f(sequence)17663 63916 y(2)p Fm(;)221 b Fo(3)p Fm(;)g Fo(6)p Fm(;)g Fo(7)p Fm(;)g Fo(8)p Fm(;)g Fo(9)p Fm(;)g Fo(12)p Fm(;)g Fo(15)p Fm(;)g Fo(18)p Fm(;)g Fo(19)p Fm(;)g(:)g(:)g(:)668 b(;)1600 66849 y Fo(whic)-36 b(h)631 b(app)36 b(eared)630 b(as)i(sequence)f(M0747)h(in)f([)p 0 1 0 0 TeXcolorcmyk(13)p (#cite.EIS) [[299 115 310 127] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])g(\()p 0 1 0 0 TeXcolorcmyk(A3605)p [[324 115 356 127] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A003605) pdfm Black 632 w(in)g([)p 0 1 0 0 TeXcolorcmyk(12)p (#cite.OEIS) [[381 115 392 127] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(]\).)1171 b(Propp)630 b(ga)-36 b(v)g(e)632 b(a)f(di\256eren)-36 b(t)1600 68454 y(\(although)636 b(equiv)-72 b(alen)-36 b(t\))636 b(de\257nition)g(in)-36 b(v)g(olving)637 b(the)f(square)g(of)h(the)f(sequence:)984 b Fl(f)p Fm(e)p Fo(\()p Fm(n)p Fo(\))p Fl(g)636 b Fo(is)g(the)1600 70059 y(unique)402 b(monotonically)i(increasing)e(sequence)g (satisfying)i Fm(e)p Fo(\()p Fm(e)p Fo(\()p Fm(n)p Fo(\)\))368 b(=)h(3)p Fm(n)403 b Fo(for)g(all)g Fm(n)369 b Fl(\270)g Fo(1.)569 b(Mic)-36 b(hael)p Black 26075 76088 a(7)p Black eop %%Page: 8 8 8 7 bop Black 0 TeXcolorgray Black Black 1600 2096 a Fo(Somos)434 b([)p 0 1 0 0 TeXcolorcmyk(14)p (#cite.Som00) [[126 698 138 710] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])g(observ)-36 b(ed)433 b(that)g(this)g(sequence)h (satis\257es)19458 5029 y Fm(e)p Fo(\(3)p Fm(n)p Fo(\))1107 b(=)g(3)p Fm(e)p Fo(\()p Fm(n)p Fo(\))p Fm(;)17206 6966 y(e)p Fo(\(3)p Fm(n)295 b Fo(+)g(1\))1107 b(=)g(2)p Fm(e)p Fo(\()p Fm(n)p Fo(\))295 b(+)g Fm(e)p Fo(\()p Fm(n)g Fo(+)g(1\))p Fm(;)17206 8904 y(e)p Fo(\(3)p Fm(n)g Fo(+)g(2\))1107 b(=)g Fm(e)p Fo(\()p Fm(n)p Fo(\))295 b(+)g(2)p Fm(e)p Fo(\()p Fm(n)g Fo(+)g(1\))p Fm(:)p Black 13516 w Fo(\(10\))p Black 1600 11837 a(An)591 b(analysis)i(similar)f(to)g(that)f(for)h Fl(f)p Fm(a)p Fo(\()p Fm(n)p Fo(\))p Fl(g)f Fo(leads)h(to)g(the)f (follo)-36 b(wing)593 b(explicit)f(form)-36 b(ula,)632 b(whic)-36 b(h)1600 13442 y(app)36 b(ears)433 b(to)h(b)36 b(e)433 b(new:)17674 15047 y Fm(e)p Fo(\(2)295 b Fl(\242)h Fo(3)21043 14499 y Fi(k)21907 15047 y Fo(+)f Fm(j)75 b Fo(\))368 b(=)h(3)26732 14499 y Fi(k)24 b Fk(+1)28799 15047 y Fo(+)295 b(2)p Fm(j)370 b Fo(+)295 b Fl(j)p Fm(j)75 b Fl(j)221 b Fm(;)p Black 13984 w Fo(\(11\))p Black 1600 17372 a(for)434 b Fm(k)414 b Fl(\270)369 b Fo(0)434 b(and)f Fl(\241)p Fo(3)11375 16890 y Fi(k)12314 17372 y Fl(\267)369 b Fm(j)444 b(<)368 b Fo(3)16728 16890 y Fi(k)17297 17372 y Fo(.)3551 18977 y(A)418 b(sequence)g(closely)h(related)f(to)g Fl(f)p Fm(e)p Fo(\()p Fm(n)p Fo(\))p Fl(g)g Fo(had)f(earlier)h(b)36 b(een)418 b(studied)e(b)-36 b(y)418 b(Arkin)g(et)g(al.)573 b([)p 0 1 0 0 TeXcolorcmyk(2)p (#cite.Ark90) [[503 546 508 558] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(,)421 b(Eq.)1600 20582 y(\(12\)].)857 b(This)527 b(is)g(the)e(sequence)i Fm(e)18374 20100 y Fj(0)18684 20582 y Fo(\()p Fm(n)p Fo(\))g(=)g Fm(e)p Fo(\()p Fm(n)p Fo(\))358 b Fl(\241)g Fm(n)527 b Fo(\()p 0 1 0 0 TeXcolorcmyk(A6166)p [[328 532 361 544] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A006166) pdfm Black(\).)858 b(Arkin)526 b(et)h(al.)857 b(giv)-36 b(e)527 b(a)g(recurrence)1600 22187 y(similar)585 b(to)g(\()p 0 1 0 0 TeXcolorcmyk(10)p (#equation.10) [[146 517 158 529] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\).)1032 b(The)584 b(sequences)h(de\257ned)e(b)-36 b(y)584 b(recurrences)g Fm(s)p Fo(\()p Fm(s)p Fo(\()p Fm(n)p Fo(\)\))625 b(=)h Fm(k)45 b(n)585 b Fo(for)g Fm(k)671 b Fl(\270)626 b Fo(4)585 b(ha)-36 b(v)g(e)1600 23792 y(recen)g(tly)434 b(b)36 b(een)433 b(studied)f(b)-36 b(y)434 b(Allouc)-36 b(he)433 b(et)h(al.)579 b([)p 0 1 0 0 TeXcolorcmyk(1)p (#cite.ARS03) [[298 503 304 515] [1 1 1 [3 3]] [0 0 1]] pdfm Black(].)3551 25397 y(The)434 b(sequence)f Fl(f)p Fm(e)p Fo(\()p Fm(n)p Fo(\))p Fl(g)h Fo(can)f(b)36 b(e)434 b(generalized)g(as) g(follo)-36 b(ws.)p Black 1600 28441 a Fn(Theorem)499 b(2.)p Black 651 w Fp(L)-66 b(et)464 b Fm(y)513 b Fp(and)464 b Fm(z)525 b Fp(b)-66 b(e)464 b(inte)-66 b(gers)463 b(of)h(opp)-66 b(osite)465 b(p)-66 b(arity)464 b(satisfying)21338 31375 y Fm(y)417 b Fl(\270)369 b Fo(2)p Fm(;)687 b(z)428 b Fl(\270)369 b Fo(2)296 b Fl(\241)f Fm(y)270 b(:)p Black 17426 w Fo(\(12\))p Black 1600 34308 a Fp(Then)376 b(ther)-66 b(e)376 b(is)h(a)g(unique)f(monotonic)-66 b(al)66 b(ly)375 b(incr)-66 b(e)g(asing)374 b(se)-66 b(quenc)g(e)375 b Fl(f)p Fm(f)142 b Fo(\()p Fm(n)p Fo(\))p Fl(g)377 b Fp(satisfying)f Fm(f)142 b Fo(\(1\))369 b(=)48298 33785 y Fk(1)p 48298 34003 471 54 v 48298 34766 a(2)48901 34308 y Fo(\()p Fm(y)147 b Fo(+)1600 35913 y Fm(z)355 b Fo(+)294 b(1\))465 b Fp(and)g Fm(f)142 b Fo(\()p Fm(f)g Fo(\()p Fm(n)p Fo(\)\))369 b(=)f Fm(y)48 b(n)296 b Fo(+)f Fm(z)524 b Fp(for)465 b Fm(n)369 b(>)g Fo(1)p Fp(.)597 b(It)464 b(is)h(given)f(by)8959 39611 y Fm(f)9963 37738 y Fe(\265)10941 39611 y Fm(c)11501 39810 y Fk(1)12027 39611 y Fm(y)12709 39062 y Fi(k)13573 39611 y Fl(\241)16180 38712 y Fm(z)p 15034 39305 2956 54 v 15034 40522 a(y)343 b Fl(\241)295 b Fo(1)18417 39611 y(+)g Fm(j)20337 37738 y Fe(\266)21684 39611 y Fo(=)369 b Fm(c)23625 39810 y Fk(2)24150 39611 y Fm(y)24832 39062 y Fi(k)24 b Fk(+1)26898 39611 y Fl(\241)29506 38712 y Fm(z)p 28359 39305 V 28359 40522 a(y)344 b Fl(\241)295 b Fo(1)31743 39611 y(+)33183 38712 y Fm(y)343 b Fo(+)295 b(1)p 33183 39305 2935 54 v 34325 40522 a(2)36250 39611 y Fm(j)370 b Fo(+)38598 38712 y Fm(y)343 b Fl(\241)295 b Fo(1)p 38598 39305 2956 54 v 39750 40522 a(2)41686 39611 y Fl(j)p Fm(j)75 b Fl(j)221 b Fm(;)p Black 5269 w Fo(\(13\))p Black 1600 43234 a Fp(for)465 b Fm(k)414 b Fl(\270)369 b Fo(0)p Fp(,)464 b(wher)-66 b(e)17079 45358 y Fl(\241)18245 44459 y Fm(y)343 b Fo(+)295 b Fm(z)355 b Fl(\241)295 b Fo(1)p 18245 45052 5222 54 v 20531 46269 a(2)23599 45358 y Fm(y)24281 44809 y Fi(k)25219 45358 y Fl(\267)369 b Fm(j)444 b(<)29116 44459 y(y)343 b Fo(+)295 b Fm(z)355 b Fl(\241)295 b Fo(1)p 29116 45052 V 31402 46269 a(2)34470 45358 y Fm(y)35152 44809 y Fi(k)1600 48022 y Fp(and)14352 50027 y Fm(c)14912 50226 y Fk(1)15807 50027 y Fo(=)17320 49129 y(\()p Fm(y)343 b Fo(+)295 b(1\)\()p Fm(y)343 b Fo(+)295 b Fm(z)354 b Fl(\241)296 b Fo(1\))p 17320 49722 10179 54 v 20101 50938 a(2\()p Fm(y)343 b Fl(\241)295 b Fo(1\))27632 50027 y Fm(;)1550 b(c)30103 50226 y Fk(2)30997 50027 y Fo(=)32511 49129 y Fm(y)343 b Fo(+)295 b Fm(z)354 b Fl(\241)296 b Fo(1)p 32511 49722 5222 54 v 33644 50938 a Fm(y)343 b Fl(\241)295 b Fo(1)38086 50027 y Fm(:)1600 54236 y Fn(Pro)42 b(of.)1301 b Fo(It)433 b(is)h(easy)g(to)g(see)g(that)f(the)g(sequence)h Fl(f)p Fm(s)p Fo(\()p Fm(n)p Fo(\))p Fl(g)f Fo(de\257ned)f(b)-36 b(y)17842 57902 y Fm(s)p Fo(\()p Fm(n)p Fo(\))802 b(=)22992 57004 y(\()p Fm(y)343 b Fo(+)295 b(1\))p Fm(n)h Fl(\241)f Fo(2)p Fm(f)142 b Fo(\()p Fm(n)p Fo(\))296 b(+)e Fm(z)p 22992 57597 11833 54 v 27431 58813 a(y)343 b Fl(\241)295 b Fo(1)p Black 48888 57902 a(\(14\))p Black 1600 61433 a(is)434 b(a)g(\\sa)-36 b(w-to)36 b(oth")434 b(sequence)g(of)g(the)f (form)8002 64367 y(0)p Fm(;)221 b Fo(1)p Fm(;)g Fo(2)p Fm(;)g Fo(3)p Fm(;)g(:)g(:)g(:)227 b(;)221 b Fo(3)p Fm(;)g Fo(2)p Fm(;)g Fo(1)p Fm(;)g Fo(0)p Fm(;)g Fo(1)p Fm(;)g Fo(2)p Fm(;)g Fo(3)p Fm(;)g Fo(4)p Fm(;)g Fo(5)p Fm(;)g Fo(6)p Fm(;)g(:)g(:)g(:)234 b(;)221 b Fo(6)p Fm(;)g Fo(5)p Fm(;)g Fo(4)p Fm(;)g Fo(3)p Fm(;)g Fo(2)p Fm(;)g Fo(1)p Fm(;)g Fo(0)p Fm(;)g Fo(1)p Fm(;)g Fo(2)p Fm(;)g Fo(3)p Fm(;)g(:)g(:)g(:)667 b(;)1600 67300 y Fo(where)433 b(the)g(0's)i(o)36 b(ccur)433 b(at)14881 70735 y Fm(n)803 b Fo(=)f(1)296 b(+)20659 69836 y(\()p Fm(y)343 b Fo(+)295 b Fm(z)354 b Fl(\241)296 b Fo(1\)\()p Fm(y)28080 69354 y Fi(k)28944 69836 y Fl(\241)f Fo(1\))p 20659 70429 10769 54 v 24566 71646 a Fm(y)343 b Fl(\241)295 b Fo(1)31995 70735 y(for)434 b Fm(k)414 b Fl(\270)369 b Fo(0)434 b Fm(:)p Black 26075 76088 a Fo(8)p Black eop %%Page: 9 9 9 8 bop Black 0 TeXcolorgray Black Black 1600 2096 a Fo(Consequen)-36 b(tly)517 b(the)g Fm(k)45 b Fo(-th)516 b(\\to)36 b(oth")517 b(0)p Fm(;)221 b Fo(1)p Fm(;)g Fo(2)p Fm(;)g Fo(3)p Fm(;)g(:)g(:)g(:)228 b(;)221 b Fo(3)p Fm(;)g Fo(2)p Fm(;)g Fo(1)p Fm(;)g Fo(0)521 b(con)-36 b(tains)516 b(\()p Fm(y)400 b Fo(+)352 b Fm(z)411 b Fl(\241)352 b Fo(1\))p Fm(y)44315 1614 y Fi(k)45401 2096 y Fo(terms)516 b(and)1600 3701 y(reac)-36 b(hes)312 b(a)g(maximal)h(v)-72 b(alue)313 b(of)g(\()p Fm(y)94 b Fo(+)47 b Fm(z)106 b Fl(\241)47 b Fo(1\))p Fm(y)22673 3219 y Fi(k)23241 3701 y Fm(=)p Fo(2.)539 b(Then)311 b(from)i(simple)f(considerations)g(of)h (symmetry)-108 b(,)1600 5306 y(for)434 b(an)-36 b(y)434 b Fm(k)414 b Fl(\270)369 b Fo(0)434 b(and)f(an)-36 b(y)434 b Fm(j)508 b Fo(suc)-36 b(h)433 b(that)g Fl(j)p Fm(j)75 b Fl(j)369 b(\267)g Fo(\()p Fm(y)343 b Fo(+)295 b Fm(z)354 b Fl(\241)296 b Fo(1\))p Fm(y)31441 4824 y Fi(k)32009 5306 y Fm(=)p Fo(2,)435 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)5400 9073 y Fm(s)6234 7200 y Fe(\265)7212 9073 y Fo(1)296 b(+)9598 8174 y(\()p Fm(y)342 b Fo(+)295 b Fm(z)355 b Fl(\241)295 b Fo(1\)\()p Fm(y)17018 7692 y Fi(k)17882 8174 y Fl(\241)g Fo(1\))p 9598 8767 10769 54 v 13504 9984 a Fm(y)343 b Fl(\241)296 b Fo(1)20794 9073 y(+)22234 8174 y(\()p Fm(y)343 b Fo(+)295 b Fm(z)354 b Fl(\241)296 b Fo(1\))p Fm(y)29149 7692 y Fi(k)p 22234 8767 7484 54 v 25651 9984 a Fo(2)30146 9073 y(+)e Fm(j)32065 7200 y Fe(\266)34067 9073 y Fo(=)36014 8174 y(\()p Fm(y)343 b Fo(+)295 b Fm(z)354 b Fl(\241)296 b Fo(1\))p 36014 8767 6233 54 v 38805 9984 a(2)42380 9073 y Fm(y)43062 8524 y Fi(k)43925 9073 y Fl(\241)g(j)p Fm(j)75 b Fl(j)433 b Fm(:)1600 12696 y Fo(In)g(other)h(w)-36 b(ords,)8450 16320 y Fm(s)9284 14447 y Fe(\265)10394 15421 y Fo(\()p Fm(y)343 b Fo(+)295 b Fm(z)355 b Fl(\241)295 b Fo(1\)\()p Fm(y)343 b Fo(+)295 b(1\))p 10394 16014 10179 54 v 13175 17231 a(2\()p Fm(y)343 b Fl(\241)295 b Fo(1\))20706 16320 y Fm(y)21388 15771 y Fi(k)22252 16320 y Fl(\241)24859 15421 y Fm(z)p 23713 16014 2956 54 v 23713 17231 a(y)343 b Fl(\241)295 b Fo(1)27096 16320 y(+)g Fm(j)29016 14447 y Fe(\266)31018 16320 y Fo(=)32965 15421 y(\()p Fm(y)343 b Fo(+)295 b Fm(z)354 b Fl(\241)295 b Fo(1\))p 32965 16014 6233 54 v 35756 17231 a(2)39330 16320 y Fm(y)40012 15771 y Fi(k)40876 16320 y Fl(\241)h(j)p Fm(j)75 b Fl(j)433 b Fm(:)p Black 4538 w Fo(\(15\))p Black 1600 20017 a(Equation)h(\()p 0 1 0 0 TeXcolorcmyk(13)p (#equation.13) [[142 537 153 549] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))g(follo)-36 b(ws)435 b(from)f(\()p 0 1 0 0 TeXcolorcmyk(14)p (#equation.14) [[232 537 244 549] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))g(and)f(\()p 0 1 0 0 TeXcolorcmyk(15)p (#equation.15) [[280 537 291 549] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\).)p 26554 19960 458 458 v 3551 21622 a(It)573 b(can)g(b)36 b(e)572 b(sho)-36 b(wn)573 b(\(w)-36 b(e)573 b(omit)g(the)g(details\))g(that)f(this)h(sequence)g(can)g(also)h(b)36 b(e)572 b(de\257ned)g(b)-36 b(y:)1600 23227 y Fm(f)142 b Fo(\(1\))449 b(=)f(\()p Fm(y)375 b Fo(+)327 b Fm(z)386 b Fo(+)327 b(1\))p Fm(=)p Fo(2,)494 b(and,)e(for)481 b Fm(n)449 b(>)g Fo(1,)492 b Fm(f)142 b Fo(\()p Fm(n)p Fo(\))481 b(is)g(the)f(smallest)h(in)-36 b(teger)480 b Fm(>)449 b(f)142 b Fo(\()p Fm(n)327 b Fl(\241)h Fo(1\))480 b(whic)-36 b(h)481 b(is)1600 24832 y(consisten)-36 b(t)504 b(with)g(the)g(condition)h(that)f(\\)p Fm(n)h Fo(is)g(a)f(mem)-36 b(b)36 b(er)504 b(of)h(the)f(sequence)h(if)g(and)f(only)h(if)g Fm(f)142 b Fo(\()p Fm(n)p Fo(\))1600 26437 y(b)36 b(elongs)434 b(to)g(the)f(set)14553 28561 y([2)p Fm(;)221 b(:)g(:)g(:)j(;)18610 27662 y Fo(1)p 18610 28255 651 54 v 18610 29472 a(2)19394 28561 y(\()p Fm(y)342 b Fo(+)295 b Fm(z)355 b Fl(\241)295 b Fo(1\)])729 b Fl([)g(f)p Fm(iy)343 b Fo(+)294 b Fm(z)429 b Fo(:)369 b Fm(i)g Fl(\270)g Fo(1)p Fl(g)221 b Fm(:)p Fo(")p Black 10642 w(\(16\))p Black 1600 31299 a(If)434 b(\()p Fm(y)343 b Fo(+)295 b Fm(z)354 b Fl(\241)296 b Fo(1\))p Fm(=)p Fo(2)370 b Fl(\267)f Fo(1,)434 b(the)f(\257rst)g(set)g (in)h(\()p 0 1 0 0 TeXcolorcmyk(16)p (#equation.16) [[276 435 287 447] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))g(is)f(to)h(b)36 b(e)433 b(omitted.)1600 34762 y Fn(Examples.)1302 b Fo(Setting)433 b Fm(y)417 b Fo(=)368 b(3,)434 b Fm(z)429 b Fo(=)368 b(0)434 b(in)g(the)f(theorem)g(pro)36 b(duces)433 b Fl(f)p Fm(e)p Fo(\()p Fm(n)p Fo(\))p Fl(g)p Fo(.)3551 36368 y(Setting)616 b Fm(y)729 b Fo(=)681 b(2,)663 b Fm(z)740 b Fo(=)681 b(1)617 b(yields)h(another)e(in)-36 b(teresting)616 b(\\mo)36 b(d)618 b(2")f(sequence.)1128 b(This)617 b(is)g(the)1600 37973 y(sequence)434 b Fl(f)p Fm(g)48 b Fo(\()p Fm(n)p Fo(\))368 b(:)h Fm(n)h Fl(\270)f Fo(1)p Fl(g)434 b Fo(that)f(b)36 b(egins)18171 40906 y(2)p Fm(;)221 b Fo(3)p Fm(;)g Fo(5)p Fm(;)g Fo(6)p Fm(;)g Fo(7)p Fm(;)g Fo(9)p Fm(;)g Fo(11)p Fm(;)g Fo(12)p Fm(;)g Fo(13)p Fm(;)g Fo(14)p Fm(;)g(:)g(:)g(:)1600 43840 y Fo(\()p 0 1 0 0 TeXcolorcmyk(A80637)p [[91 322 129 334] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A080637) pdfm Black(\).)580 b(It)433 b(has)h(the)f(follo)-36 b(wing)436 b(prop)36 b(erties:)3551 45445 y(\(i\))457 b(By)h(de\257nition,)k(this) 457 b(is)g(the)g(unique)g(monotonically)h(increasing)g(sequence)f Fl(f)p Fm(g)48 b Fo(\()p Fm(n)p Fo(\))p Fl(g)457 b Fo(satis-)1600 47050 y(fying)435 b Fm(g)48 b Fo(\(1\))368 b(=)h(2,)434 b Fm(g)48 b Fo(\()p Fm(g)g Fo(\()p Fm(n)p Fo(\)\))367 b(=)i(2)p Fm(n)296 b Fo(+)f(1)434 b(for)g Fm(n)369 b Fl(\270)g Fo(2.)3551 48655 y(\(ii\))434 b Fm(n)g Fo(is)g(in)f(the)g (sequence)h(if)g(and)f(only)h(if)g Fm(g)48 b Fo(\()p Fm(n)p Fo(\))434 b(is)f(an)h(o)36 b(dd)433 b(n)-36 b(um)g(b)36 b(er)432 b Fl(\270)369 b Fo(3.)3551 50260 y(\(iii\))434 b(The)g(sequence)f(of)h(\257rst)f(di\256erences)g(is)h(\()p 0 1 0 0 TeXcolorcmyk(A79882)p [[310 265 348 277] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A079882) pdfm Black(\):)16606 53193 y(1)p Fm(;)221 b Fo(2)p Fm(;)g Fo(1)19720 52645 y Fk(2)20249 53193 y Fm(;)g Fo(2)21481 52645 y Fk(2)22007 53193 y Fm(;)g Fo(1)23239 52645 y Fk(4)23766 53193 y Fm(;)g Fo(2)24998 52645 y Fk(4)25525 53193 y Fm(;)g Fo(1)26757 52645 y Fk(8)27284 53193 y Fm(;)g Fo(2)28516 52645 y Fk(8)29043 53193 y Fm(;)g Fo(1)30275 52645 y Fk(16)31272 53193 y Fm(;)g Fo(2)32504 52645 y Fk(16)33501 53193 y Fm(;)g(:)g(:)g(:)j(:)3551 56127 y Fo(\(iv\))14082 58324 y Fm(g)48 b Fo(\(3)294 b Fl(\242)i Fo(2)17518 57776 y Fi(k)18382 58324 y Fl(\241)f Fo(1)h(+)f Fm(j)75 b Fo(\))368 b(=)h(2)295 b Fl(\242)h Fo(2)27091 57776 y Fi(k)24 b Fk(+1)29157 58324 y Fl(\241)296 b Fo(1)f(+)32871 57426 y(3)p 32871 58019 V 32871 59235 a(2)33654 58324 y Fm(j)370 b Fo(+)36002 57426 y(1)p 36002 58019 V 36002 59235 a(2)36785 58324 y Fl(j)p Fm(j)75 b Fl(j)221 b Fm(;)1600 61163 y Fo(for)434 b Fm(k)414 b Fl(\270)369 b Fo(0,)434 b Fl(\241)p Fo(2)9207 60681 y Fi(k)10146 61163 y Fl(\267)369 b Fm(j)444 b(<)368 b Fo(2)14560 60681 y Fi(k)15563 61163 y Fo(\(from)434 b(\()p 0 1 0 0 TeXcolorcmyk(13)p (#equation.13) [[249 167 261 179] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\)\).)3551 62768 y(\(v\))427 b Fm(g)48 b Fo(\(2)p Fm(n)p Fo(\))368 b(=)h Fm(g)48 b Fo(\()p Fm(n)p Fo(\))280 b(+)h Fm(g)48 b Fo(\()p Fm(n)281 b Fl(\241)g Fo(1\))g(+)f(1,)429 b Fm(g)48 b Fo(\(2)p Fm(n)281 b Fo(+)f(1\))369 b(=)g(2)p Fm(g)48 b Fo(\()p Fm(n)p Fo(\))281 b(+)f(1,)429 b(for)e Fm(n)369 b Fl(\270)g Fo(1)427 b(\(taking)g Fm(g)48 b Fo(\(0\))369 b(=)f(0\).)3551 64373 y(\(vi\))416 b(The)g(original)i (sequence)e Fl(f)p Fm(a)p Fo(\()p Fm(n)p Fo(\))p Fl(g)g Fo(satis\257es)g Fm(a)p Fo(\(3)p Fm(n)p Fo(\))369 b(=)g(3)p Fm(g)48 b Fo(\()p Fm(n)p Fo(\))p Fm(;)221 b(a)p Fo(\(3)p Fm(n)260 b Fo(+)f(1\))369 b(=)g(2)p Fm(g)48 b Fo(\()p Fm(n)p Fo(\))259 b(+)g Fm(g)48 b Fo(\()p Fm(n)259 b Fo(+)1600 65978 y(1\))p Fm(;)221 b(a)p Fo(\(3)p Fm(n)296 b Fo(+)f(2\))369 b(=)g Fm(g)48 b Fo(\()p Fm(n)p Fo(\))295 b(+)f(2)p Fm(g)48 b Fo(\()p Fm(n)296 b Fo(+)e(1\),)434 b(for)g Fm(n)370 b Fl(\270)f Fo(1.)3551 67583 y(\(vii\))434 b(The)g(\\lying)h(v)-36 b(ersion")434 b(of)g(Section)p 0 1 0 0 TeXcolorcmyk 434 w(3)p (#section.3) [[288 109 294 121] [1 1 1 [3 3]] [0 0 1]] pdfm Black 434 w(is)f(giv)-36 b(en)435 b(b)-36 b(y)433 b Fm(d)p Fo(\()p Fm(n)p Fo(\))369 b(=)g Fm(g)48 b Fo(\()p Fm(n)294 b Fo(+)h(1\))h Fl(\241)f Fo(1)434 b(for)g Fm(n)369 b Fl(\270)g Fo(1.)3551 69188 y(\(viii\))610 b(Let)f Fm(g)10119 68706 y Fj(0)10429 69188 y Fo(\()p Fm(n)p Fo(\))668 b(=)g Fm(g)48 b Fo(\()p Fm(n)p Fo(\))414 b(+)h(1.)1106 b(The)609 b(sequence)g Fl(f)p Fm(g)30732 68706 y Fj(0)31043 69188 y Fo(\()p Fm(n)p Fo(\))668 b(:)g Fm(n)h Fl(\270)f Fo(2)p Fl(g)610 b Fo(w)-36 b(as)610 b(apparen)-36 b(tly)609 b(\257rst)1600 70793 y(disco)-36 b(v)g(ered)549 b(b)-36 b(y)549 b(C.)h(L.)f(Mallo)-36 b(ws,)580 b(and)548 b(is)i(sequence)f (M2317)h(in)f([)p 0 1 0 0 TeXcolorcmyk(13)p (#cite.EIS) [[387 80 398 92] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])h(\()p 0 1 0 0 TeXcolorcmyk(A7378)p [[411 80 443 92] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A007378) pdfm Black 550 w(in)f([)p 0 1 0 0 TeXcolorcmyk(12)p (#cite.OEIS) [[466 80 478 92] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(]\).)925 b(This)550 b(is)1600 72398 y(the)461 b(unique)g(monotonically)i(increasing)f(sequence)f(satisfying)i Fm(g)34586 71916 y Fj(0)34896 72398 y Fo(\()p Fm(g)36073 71916 y Fj(0)36383 72398 y Fo(\()p Fm(n)p Fo(\)\))416 b(=)g(2)p Fm(n)p Fo(.)663 b(An)461 b(alternativ)-36 b(e)p Black 26075 76088 a(9)p Black eop %%Page: 10 10 10 9 bop Black 0 TeXcolorgray Black Black 1600 2096 a Fo(description)545 b(is:)804 b Fm(g)11077 1614 y Fj(0)11387 2096 y Fo(\()p Fm(n)p Fo(\))545 b(\(for)i Fm(n)560 b Fl(\270)g Fo(2\))546 b(is)g(the)g(smallest)g(p)36 b(ositiv)-36 b(e)547 b(in)-36 b(teger)546 b Fm(>)559 b(g)41421 1614 y Fj(0)41731 2096 y Fo(\()p Fm(n)372 b Fl(\241)g Fo(1\))546 b(whic)-36 b(h)546 b(is)1600 3701 y(consisten)-36 b(t)433 b(with)h(the)f(condition)g(that)5910 6412 y(\\)p Fm(n)h Fo(is)g(a)g(mem)-36 b(b)36 b(er)433 b(of)h(the)f(sequence)h(if)g(and)f (only)h(if)g Fm(g)32698 5930 y Fj(0)33008 6412 y Fo(\()p Fm(n)p Fo(\))g(is)g(an)f(ev)-36 b(en)434 b(n)-36 b(um)g(b)36 b(er)432 b Fl(\270)369 b Fo(4")q Fm(:)p Black -216 w Fo(\(17\))p Black 1600 9124 a(Note)390 b(that,)399 b(although)390 b(\()p 0 1 0 0 TeXcolorcmyk(8)p (#equation.8) [[196 635 201 647] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))h(and)e(\()p 0 1 0 0 TeXcolorcmyk(17)p (#equation.17) [[236 635 248 647] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))i(are)f(similar,)400 b(the)390 b(resulting)g(sequences)g Fl(f)p Fm(c)p Fo(\()p Fm(n)p Fo(\))p Fl(g)h Fo(and)f Fl(f)p Fm(g)48438 8642 y Fj(0)48748 9124 y Fo(\()p Fm(n)p Fo(\))p Fl(g)1600 10729 y Fo(are)598 b(quite)g(di\256eren)-36 b(t.)1071 b Fm(g)14185 10247 y Fj(0)15093 10729 y Fo(is)598 b(not)g(directly)g(co)-36 b(v)g(ered)598 b(b)-36 b(y)598 b(Theorem)p 0 1 0 0 TeXcolorcmyk 598 w(2)p (#theorem.2) [[400 620 405 632] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)640 b(and)597 b(w)-36 b(e)598 b(admit)g(that)g(w)-36 b(e)1600 12334 y(ha)g(v)g(e)390 b(not)g(b)36 b(een)390 b(able)h(to)f(iden)-36 b(tify)391 b(the)f(largest)h(family)h(of)f (sequences)f(whic)-36 b(h)390 b(can)g(b)36 b(e)390 b(describ)36 b(ed)390 b(b)-36 b(y)1600 13939 y(form)g(ulae)434 b(lik)-36 b(e)435 b(\()p 0 1 0 0 TeXcolorcmyk(3)p (#equation.3) [[161 592 166 604] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\),)f(\()p 0 1 0 0 TeXcolorcmyk(7)p (#equation.7) [[183 592 189 604] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\),)f(\()p 0 1 0 0 TeXcolorcmyk(11)p (#equation.11) [[205 592 216 604] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\),)h(\()p 0 1 0 0 TeXcolorcmyk(13)p (#equation.13) [[233 592 244 604] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\).)3551 15544 y(The)577 b(sequence)h Fl(f)p Fm(h)p Fo(\()p Fm(n)p Fo(\))p Fl(g)f Fo(de\257ned)f(b)-36 b(y:)865 b Fm(h)p Fo(\(1\))614 b(=)f(2,)h(and,)f(for)578 b Fm(n)614 b(>)f Fo(1,)h Fm(h)p Fo(\()p Fm(n)p Fo(\))577 b(is)g(the)g(smallest) 1600 17149 y(p)36 b(ositiv)-36 b(e)450 b(in)-36 b(teger)449 b Fm(>)396 b(h)p Fo(\()p Fm(n)305 b Fl(\241)h Fo(1\))450 b(whic)-36 b(h)449 b(is)h(consisten)-36 b(t)448 b(with)i(the)e (condition)i(that)e(\\)p Fm(n)i Fo(is)g(a)g(mem)-36 b(b)36 b(er)1600 18755 y(of)434 b(the)f(sequence)h(if)g(and)f(only)h(if)h Fm(h)p Fo(\()p Fm(n)p Fo(\))e(is)h(a)g(m)-36 b(ultiple)433 b(of)h(6":)17444 21466 y(2)p Fm(;)221 b Fo(6)p Fm(;)g Fo(7)p Fm(;)g Fo(8)p Fm(;)g Fo(9)p Fm(;)g Fo(12)p Fm(;)g Fo(18)p Fm(;)g Fo(24)p Fm(;)g Fo(30)p Fm(;)g Fo(31)p Fm(;)g(:)g(:)g(:)456 b(;)1600 24178 y Fo(\()p 0 1 0 0 TeXcolorcmyk(A80780)p [[91 499 129 511] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A080780) pdfm Black(\),)422 b(sho)-36 b(ws)417 b(that)f(suc)-36 b(h)416 b(simple)g(rules)h(do)g(not)f(hold)g(in)h(general.)573 b(W)-108 b(e)417 b(can)g(c)-36 b(haracterize)416 b(the)1600 25783 y(sequence)512 b(of)h(\257rst)e(di\256erences)h(in)f(a)i(manner)e (similar)i(to)f(\()p 0 1 0 0 TeXcolorcmyk(5)p (#equation.5) [[359 485 365 497] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\),)532 b(\()p 0 1 0 0 TeXcolorcmyk(6)p (#equation.6) [[382 485 388 497] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\):)736 b(the)511 b(alphab)36 b(et)512 b(is)g(no)-36 b(w)513 b Fl(A)e Fo(=)1600 27388 y Fl(f)p Fo(1)p Fm(;)221 b Fo(2)p Fm(;)g(:)g(:)g(:)k(;)c Fo(6)p Fl(g)p Fo(,)340 b(and)313 b(w)-36 b(e)315 b(de\257ne)e(a)i(mapping)e Fm(\265)351 b Fo(from)315 b Fl(A)28080 26906 y Fj(\244)28919 27388 y Fo(to)g Fl(A)31451 26906 y Fj(\244)32290 27388 y Fo(b)-36 b(y)314 b(the)g(rules)g Fm(\265)36 b Fo(\()p Fm(i)p Fo(\))369 b(=)f(1)p Fm(;)221 b Fo(1)p Fm(;)g(:)g(:)g(:)k(;)c Fo(1)p Fm(;)g Fo(7)51 b Fl(\241)g Fm(i)1600 28993 y Fo(\(with)478 b Fm(i)325 b Fl(\241)h Fo(1)479 b(1's)g(follo)-36 b(w)g(ed)480 b(b)-36 b(y)478 b(7)326 b Fl(\241)f Fm(i)p Fo(\),)490 b(for)479 b Fm(i)444 b Fo(=)h(1)p Fm(;)221 b(:)g(:)g(:)j(;)d Fo(6.)714 b(Then)478 b(the)f(sequence)i(of)g(di\256erences)e(of)1600 30598 y Fl(f)p Fm(h)p Fo(\()p Fm(n)p Fo(\))p Fl(g)499 b Fo(is)h Fm(S)8138 30797 y Fk(0)8664 30598 y Fm(;)221 b(S)10046 30797 y Fk(1)10572 30598 y Fm(;)g(S)11954 30797 y Fk(2)12481 30598 y Fm(;)g(:)g(:)g(:)r Fo(,)516 b(where)499 b Fm(S)20090 30797 y Fk(0)21097 30598 y Fo(=)480 b Fl(f)p Fo(4)p Fl(g)p Fm(;)221 b(S)25949 30797 y Fi(k)24 b Fk(+1)28203 30598 y Fo(=)480 b Fm(\265)36 b Fo(\()p Fm(S)31643 30797 y Fi(k)32212 30598 y Fo(\))499 b(for)h Fm(k)526 b Fl(\270)481 b Fo(0.)776 b(Ho)-36 b(w)g(ev)g(er,)517 b(it)499 b(app)36 b(ears)1600 32203 y(that)433 b(no)h(form)-36 b(ula)434 b(similar)g(to)g(\()p 0 1 0 0 TeXcolorcmyk(3)p (#equation.3) [[228 427 234 439] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))f(holds)h(for)g Fm(h)p Fo(\()p Fm(n)p Fo(\).)3551 33808 y(W)-108 b(e)539 b(end)g(this)f(\\mo)36 b(d)540 b(m")f(section)h(with)f(t)-36 b(w)g(o)539 b(in)-36 b(teresting)539 b(\\mo)36 b(d)540 b(4")f(sequences.)896 b(The)539 b(ev)-36 b(en)1600 35413 y(n)g(um)g(b)36 b(ers)592 b(satisfy)k Fm(s)p Fo(\()p Fm(s)p Fo(\()p Fm(n)p Fo(\)\))640 b(=)i(4)p Fm(n)p Fo(,)635 b(and)593 b(the)g(o)36 b(dd)594 b(n)-36 b(um)g(b)36 b(ers)592 b(satisfy)k Fm(s)p Fo(\()p Fm(s)p Fo(\()p Fm(n)p Fo(\)\))640 b(=)i(4)p Fm(n)405 b Fo(+)f(3.)1059 b(But)1600 37018 y(there)565 b(are)h(lexicographically)i(earlier)e (sequences)g(with)g(the)f(same)g(prop)36 b(erties.)974 b(The)566 b(\\pseudo-)1600 38623 y(ev)-36 b(en)585 b(n)-36 b(um)g(b)36 b(ers")584 b Fl(f)p Fm(i)p Fo(\()p Fm(n)p Fo(\))626 b(:)h Fm(n)g Fl(\270)g Fo(0)p Fl(g)585 b Fo(are)g(de\257ned)f (b)-36 b(y)585 b(the)f(prop)36 b(ert)-36 b(y)584 b(that)h Fm(i)p Fo(\()p Fm(n)p Fo(\))g(is)g(the)f(smallest)1600 40229 y(nonnegativ)-36 b(e)434 b(in)-36 b(teger)433 b Fm(>)369 b(i)p Fo(\()p Fm(n)295 b Fl(\241)g Fo(1\))434 b(and)f(satisfying)i Fm(i)p Fo(\()p Fm(i)p Fo(\()p Fm(n)p Fo(\)\))368 b(=)h(4)p Fm(n)434 b Fo(\()p 0 1 0 0 TeXcolorcmyk(A80588)p [[392 355 430 367] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A080588) pdfm Black(\):)17846 42940 y(0)p Fm(;)221 b Fo(2)p Fm(;)g Fo(4)p Fm(;)g Fo(5)p Fm(;)g Fo(8)p Fm(;)g Fo(12)p Fm(;)g Fo(13)p Fm(;)g Fo(14)p Fm(;)g Fo(16)p Fm(;)g Fo(17)p Fm(;)g(:)g(:)g(:)1600 45652 y Fo(W)-108 b(e)434 b(analyze)g(this)g (sequence)f(b)-36 b(y)434 b(describing)f(the)g(sequence)g(of)i(\257rst) e(di\256erences,)g(whic)-36 b(h)433 b(are)15772 48363 y(2)p Fm(;)221 b Fo(2)p Fm(;)g Fo(1)p Fm(;)g Fo(3)p Fm(;)g Fo(4)p Fm(;)g Fo(1)p Fm(;)g Fo(1)p Fm(;)g Fo(2)p Fm(;)g Fo(1)p Fm(;)g Fo(1)p Fm(;)g Fo(1)p Fm(;)g Fo(1)p Fm(;)g Fo(4)p Fm(;)g Fo(4)p Fm(;)g Fo(1)p Fm(;)g Fo(3)p Fm(;)g(:)g(:)g(:)1600 51075 y Fo(After)434 b(the)f(initial)h(2,)g(2,)h(1,)f(this)f(breaks)h (up)f(in)-36 b(to)433 b(segmen)-36 b(ts)433 b(of)i(the)e(form)23352 53786 y(3)h Fm(S)25236 53985 y Fi(k)26238 53786 y Fo(2)g Fm(T)28084 53985 y Fi(k)29087 53786 y Fm(;)1600 56498 y Fo(where)f Fm(T)6119 56697 y Fi(k)7122 56498 y Fo(is)h(the)f(rev)-36 b(ersal)434 b(of)17708 59210 y(1)18358 58661 y Fk(1)19318 59210 y Fo(4)19968 58661 y Fk(2)20927 59210 y Fo(1)21577 58661 y Fk(4)22537 59210 y Fo(4)23187 58661 y Fk(8)24147 59210 y Fo(1)24797 58661 y Fk(16)26227 59210 y Fo(4)26877 58661 y Fk(32)28095 59210 y Fl(\242)221 b(\242)g(\242)h Fo(4)30516 58661 y Fk(2)30986 58349 y Fd(2)p Fc(k)14 b Fb(\241)p Fd(1)33002 59210 y Fo(1)33652 58661 y Fk(2)34122 58349 y Fd(2)p Fc(k)1600 61921 y Fo(and)433 b Fm(S)4929 62120 y Fi(k)5931 61921 y Fo(is)h(the)f(rev)-36 b(ersal)435 b(of)16894 64633 y(1)17544 64084 y Fk(2)18504 64633 y Fo(4)19154 64084 y Fk(1)20113 64633 y Fo(1)20763 64084 y Fk(8)21723 64633 y Fo(4)22373 64084 y Fk(4)23333 64633 y Fo(1)23983 64084 y Fk(32)25413 64633 y Fo(4)26063 64084 y Fk(16)27281 64633 y Fl(\242)221 b(\242)g(\242)h Fo(1)29702 64084 y Fk(2)30172 63772 y Fd(2)p Fc(k)14 b Fb(\241)p Fd(1)32188 64633 y Fo(4)32838 64084 y Fk(2)33308 63772 y Fd(2)p Fc(k)g Fb(\241)p Fd(2)35545 64633 y Fm(:)1600 67344 y Fo(The)434 b(\\pseudo-o)36 b(dd)432 b(n)-36 b(um)g(b)36 b(ers",)433 b Fm(i)18534 66862 y Fj(0)18844 67344 y Fo(\()p Fm(n)p Fo(\),)h(are)f(similarly)j(de\257ned)c(b)-36 b(y)433 b Fm(i)35643 66862 y Fj(0)35954 67344 y Fo(\()p Fm(i)36904 66862 y Fj(0)37214 67344 y Fo(\()p Fm(n)p Fo(\)\))368 b(=)h(4)p Fm(n)296 b Fo(+)f(3:)17521 70056 y(1)p Fm(;)221 b Fo(3)p Fm(;)g Fo(4)p Fm(;)g Fo(7)p Fm(;)g Fo(11)p Fm(;)g Fo(12)p Fm(;)g Fo(13)p Fm(;)g Fo(15)p Fm(;)g Fo(16)p Fm(;)g Fo(17)p Fm(;)g(:)g(:)g(:)1600 72767 y Fo(\()p 0 1 0 0 TeXcolorcmyk(A80591)p [[91 62 129 74] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A080591) pdfm Black(\),)435 b(and)e(satisfy)i Fm(i)14667 72285 y Fj(0)14977 72767 y Fo(\()p Fm(n)p Fo(\))369 b(=)g Fm(i)p Fo(\()p Fm(n)295 b Fo(+)g(1\))g Fl(\241)h Fo(1.)p Black 25750 76088 a(10)p Black eop %%Page: 11 11 11 10 bop Black 0 TeXcolorgray Black Black 1600 2096 a Fq(5.)1614 b(The)716 b(Aronson)g(transform)1600 5016 y Fo(A)392 b(far-reac)-36 b(hing)392 b(generalization)i(of)f(b)36 b(oth)391 b(the)h(original)h(sequence)g(and)e(the)h(\\mo)36 b(d)392 b(m")h(extensions)1600 6621 y(of)353 b(the)e(previous)h (section)g(is)g(obtained)g(if)g(w)-36 b(e)353 b(replace)f(\\o)36 b(dd)351 b(n)-36 b(um)g(b)36 b(er")351 b(in)h(the)f(de\257nition)g(of)i Fl(f)p Fm(a)p Fo(\()p Fm(n)p Fo(\))p Fl(g)1600 8226 y Fo(b)-36 b(y)433 b(\\mem)-36 b(b)36 b(er)434 b(of)g Fa(\257)46 b Fo(",)434 b(where)f Fa(\257)479 b Fo(is)434 b(some)g(\257xed)f (sequence.)3551 9832 y(More)560 b(precisely)-108 b(,)592 b(let)560 b(us)g(\257x)g(a)g(starting)g(p)36 b(oin)-36 b(t)560 b Fm(n)29020 10031 y Fk(0)29546 9832 y Fo(,)592 b(whic)-36 b(h)559 b(will)j(normally)e(b)36 b(e)560 b(0)g(or)h(1.)957 b(Let)1600 11437 y Fa(\257)476 b Fo(=)430 b Fl(f)p Fm(\257)74 b Fo(\()p Fm(n)p Fo(\))430 b(:)h Fm(n)f Fl(\270)h Fm(n)12322 11636 y Fk(0)12848 11437 y Fl(g)470 b Fo(b)36 b(e)469 b(an)h(in\257nite)f(monotonically)i(increasing)f(sequence)f(of)i(in)-36 b(tegers)469 b Fl(\270)431 b Fm(n)50674 11636 y Fk(0)1600 13042 y Fo(with)548 b(the)f(prop)36 b(ert)-36 b(y)547 b(that)g(its)g(complemen)-36 b(t)547 b(\(the)g(n)-36 b(um)g(b)36 b(ers)546 b Fl(\270)563 b Fm(n)35283 13241 y Fk(0)36357 13042 y Fo(that)547 b(are)h(not)f(in)g Fa(\257)46 b Fo(\))547 b(is)h(also)1600 14647 y(in\257nite.)573 b(Then)420 b(the)g(sequence)h Fa(\256)369 b Fo(=)g Fl(f)p Fm(\256)8 b Fo(\()p Fm(n)p Fo(\))370 b(:)f Fm(n)h Fl(\270)f Fm(n)27968 14846 y Fk(0)28494 14647 y Fl(g)421 b Fo(giv)-36 b(en)421 b(b)-36 b(y:)572 b Fm(\256)8 b Fo(\()p Fm(n)p Fo(\))421 b(is)g(the)f(smallest)h(p)36 b(ositiv)-36 b(e)1600 16252 y(in)g(teger)433 b Fm(>)369 b(\256)8 b Fo(\()p Fm(n)296 b Fl(\241)f Fo(1\))434 b(whic)-36 b(h)434 b(is)f(consisten)-36 b(t)434 b(with)f(the)g(condition)h(that)15751 18878 y(\\n)f(is)h(in)g Fa(\256)g Fo(if)g(and)f(only)h(if)h Fm(\256)8 b Fo(\()p Fm(n)p Fo(\))434 b(is)g(in)f Fa(\257)46 b Fo(")1600 21505 y(is)434 b(called)g(the)f Fp(A)-33 b(r)-66 b(onson)464 b(tr)-66 b(ansform)432 b Fo(of)i Fa(\257)46 b Fo(.)p Black 1600 24198 a Fn(Theorem)499 b(3.)p Black 651 w Fp(The)465 b(A)-33 b(r)-66 b(onson)463 b(tr)-66 b(ansform)464 b(exists)g(and)h(is)g(unique.)1600 27616 y Fn(Pro)42 b(of.)1301 b Fo(F)-108 b(or)597 b(ease)i(of)g(discussion)f(let)g(us)g (call)h(the)e(n)-36 b(um)g(b)36 b(ers)597 b(in)h Fa(\257)643 b Fo(\\hot",)d(and)598 b(those)g(in)g(its)1600 29222 y(complemen)-36 b(t)557 b(\\cold.")951 b(W)-108 b(e)557 b(will)i(sp)36 b(ecify)558 b(the)f(transform)h Fa(\256)p Fo(,)589 b(lea)-36 b(ving)559 b(to)e(the)g(reader)g(the)g(easy)1600 30827 y(v)-36 b(eri\257cation)467 b(that)f(this)g(has)g(the)g(desired)g (prop)36 b(erties,)474 b(in)467 b(particular)f(that)g(there)g(are)g(no) h(con)-36 b(tra-)1600 32432 y(dictions.)3551 34037 y(The)552 b(pro)36 b(of)553 b(is)f(b)-36 b(y)552 b(induction.)933 b(First)552 b(w)-36 b(e)553 b(consider)f(the)f(initial)i(term)f Fm(\256)8 b Fo(\()p Fm(n)41933 34236 y Fk(0)42460 34037 y Fo(\).)933 b(If)553 b Fm(n)46455 34236 y Fk(0)47533 34037 y Fo(is)g(hot,)1600 35642 y Fm(\256)8 b Fo(\()p Fm(n)3717 35841 y Fk(0)4244 35642 y Fo(\))369 b(=)f Fm(n)7275 35841 y Fk(0)7801 35642 y Fo(.)579 b(If)434 b Fm(n)10817 35841 y Fk(0)11777 35642 y Fo(is)g(cold,)g Fm(\256)8 b Fo(\()p Fm(n)18309 35841 y Fk(0)18836 35642 y Fo(\))368 b(=)802 b(smallest)435 b(cold)f(n)-36 b(um)g(b)36 b(er)801 b Fl(\270)369 b Fm(n)36551 35841 y Fk(0)37372 35642 y Fo(+)295 b(1.)3551 37247 y(F)-108 b(or)433 b(the)g(induction)g(step,)g (supp)36 b(ose)433 b Fm(\256)8 b Fo(\()p Fm(n)p Fo(\))370 b(=)f Fm(k)478 b Fo(for)434 b Fm(n)370 b(>)e(n)32725 37446 y Fk(0)33251 37247 y Fo(.)3551 38852 y(Case)600 b(\(i\),)641 b Fm(k)697 b Fo(=)651 b Fm(n)p Fo(.)1077 b(If)600 b Fm(n)408 b Fo(+)g(1)600 b(is)g(hot)f(then)g Fm(\256)8 b Fo(\()p Fm(n)409 b Fo(+)e(1\))652 b(=)f Fm(n)409 b Fo(+)e(1.)1077 b(If)600 b Fm(n)409 b Fo(+)e(1)600 b(is)g(cold)g(then) 1600 40457 y Fm(\256)8 b Fo(\()p Fm(n)296 b Fo(+)f(1\))369 b(=)802 b(smallest)435 b(cold)e(n)-36 b(um)g(b)36 b(er)801 b Fl(\270)370 b Fm(n)295 b Fo(+)g(2.)3551 42062 y(Case)500 b(\(ii\),)517 b Fm(k)527 b(>)481 b(n)p Fo(.)777 b(If)500 b Fm(k)526 b Fo(=)481 b Fm(n)341 b Fo(+)f(1)500 b(then)e Fm(\256)8 b Fo(\()p Fm(n)341 b Fo(+)f(1\))482 b(=)914 b(smallest)435 b(hot)e(n)-36 b(um)g(b)36 b(er)913 b Fl(\270)482 b Fm(n)341 b Fo(+)e(2.)777 b(If)1600 43667 y Fm(k)414 b(>)369 b(n)282 b Fo(+)f(1)427 b(then)f(if)i Fm(n)282 b Fo(+)f(1)427 b(is)g(hot,)i Fm(\256)8 b Fo(\()p Fm(n)282 b Fo(+)f(1\))370 b(=)802 b(smallest)434 b(hot)f(n)-36 b(um)g(b)36 b(er)801 b Fl(\270)369 b Fm(k)327 b Fo(+)281 b(1,)429 b(while)f(if)f Fm(n)282 b Fo(+)g(1)1600 45272 y(is)434 b(cold,)g Fm(\256)8 b Fo(\()p Fm(n)296 b Fo(+)f(1\))369 b(=)802 b(smallest)435 b(cold)e(n)-36 b(um)g(b)36 b(er)801 b Fl(\270)370 b Fm(k)340 b Fo(+)295 b(1.)p 32538 45215 458 458 v 3551 46878 a(In)521 b(certain)g(cases)h(it)g(ma)-36 b(y)522 b(b)36 b(e)521 b(appropriate)g(to)g(sp)36 b(ecify)523 b(some)e(initial)i(terms)e(in)g Fa(\256)h Fo(to)f(get)h(it)1600 48483 y(started)433 b(prop)36 b(erly)-108 b(.)1600 51901 y Fn(Examples.)1302 b Fo(Of)567 b(course)g(taking)g Fa(\257)613 b Fo(to)567 b(b)36 b(e)567 b(the)f(o)36 b(dd)567 b(n)-36 b(um)g(b)36 b(ers)565 b(\(with)i Fm(n)39753 52100 y Fk(0)40875 51901 y Fo(=)596 b(1\))567 b(leads)g(to)g(our)1600 53506 y(original)418 b(sequence)f Fl(f)p Fm(a)p Fo(\()p Fm(n)p Fo(\))p Fl(g)p Fo(,)j(and)d(the)f(ev)-36 b(en)417 b(n)-36 b(um)g(b)36 b(ers)415 b(\(with)i Fm(n)33373 53705 y Fk(0)34268 53506 y Fo(=)369 b(0\))417 b(lead)g(to)g Fl(f)p Fm(c)p Fo(\()p Fm(n)p Fo(\))p Fl(g)g Fo(of)g(Section)p 0 1 0 0 TeXcolorcmyk 1600 55111 a(3)p (#section.3) [[86 221 92 233] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)3551 56716 y(If)402 b(w)-36 b(e)401 b(tak)-36 b(e)402 b Fa(\257)446 b Fo(to)401 b(b)36 b(e)401 b(the)g(triangular)g (n)-36 b(um)g(b)36 b(ers)399 b(w)-36 b(e)402 b(get)f(1,)408 b(4,)g(5,)g(6,)g(10,)h(15,)f(16,)h(17,)f(18,)h(21,)f Fm(:)221 b(:)g(:)1600 58322 y Fo(\()p 0 1 0 0 TeXcolorcmyk(A79257)p [[91 192 129 204] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A079257) pdfm Black(\);)393 b(the)369 b(squares)h(giv)-36 b(e)371 b(1,)384 b(3,)f(4,)g(9,)h(10,)f(11,)h(12,)g(13,)f(16,)h(25,)f Fm(:)221 b(:)g(:)372 b Fo(\()p 0 1 0 0 TeXcolorcmyk(A79258)p [[406 192 445 204] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A079258) pdfm Black(\);)393 b(the)369 b(primes)h(giv)-36 b(e)1600 59927 y(4,)415 b(6,)h(8,)f(11,)h(12,)f(13,)h(14,)f(17,)h(18,)g(20,)f Fm(:)221 b(:)g(:)412 b Fo(\()p 0 1 0 0 TeXcolorcmyk(A79254)p [[278 178 316 190] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A079254) pdfm Black(\);)419 b(and)410 b(the)f(lo)-36 b(w)g(er)411 b(Wytho\256)f (sequence)g(\()p 0 1 0 0 TeXcolorcmyk(A201)p [[499 178 525 190] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000201) pdfm Black(\),)1600 61532 y(in)433 b(whic)-36 b(h)434 b(the)f Fm(n)9855 61050 y Fk(th)11233 61532 y Fo(term)g(is)h Fl(b)p Fm(n\301)p Fl(c)p Fo(,)h(giv)-36 b(es)434 b(1,)g(5,)h(7,)f(10,)g (11,)h(13,)f(14,)h(15,)f(18,)h(19,)f Fm(:)221 b(:)g(:)436 b Fo(\()p 0 1 0 0 TeXcolorcmyk(A80760)p [[466 163 504 175] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A080760) pdfm Black(\).)3551 63137 y(T)-108 b(aking)496 b(the)e(Aronson)g(transform)h (of)g Fl(f)p Fm(a)p Fo(\()p Fm(n)p Fo(\))p Fl(g)g Fo(itself)g(w)-36 b(e)495 b(get)g(1,)510 b(3,)h(4,)f(6,)h(10,)g(11,)f(12,)h(14,)g(22,) 1600 64742 y(23,)434 b Fm(:)221 b(:)g(:)436 b Fo(\()p 0 1 0 0 TeXcolorcmyk(A79325)p [[127 134 165 146] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A079325) pdfm Black(\).)579 b([)p 0 1 0 0 TeXcolorcmyk(12)p (#cite.OEIS) [[182 134 193 146] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])434 b(con)-36 b(tains)434 b(sev)-36 b(eral)434 b(other)f(examples.)3551 66347 y(The)580 b(in)-36 b(v)g(erse)581 b(transform)g(ma)-36 b(y)581 b(b)36 b(e)580 b(de\257ned)f(in)h(a)h (similar)g(w)-36 b(a)g(y)-108 b(.)1020 b(Giv)-36 b(en)580 b(an)h(in\257nite)e(mono-)1600 67952 y(tonically)594 b(increasing)f(sequence)g Fa(\256)640 b Fo(=)g Fl(f)p Fm(\256)8 b Fo(\()p Fm(n)p Fo(\))641 b(:)f Fm(n)g Fl(\270)g Fm(n)30952 68151 y Fk(0)31478 67952 y Fl(g)593 b Fo(of)h(n)-36 b(um)g(b)36 b(ers)591 b Fl(\270)640 b Fm(n)42230 68151 y Fk(0)42756 67952 y Fo(,)633 b(suc)-36 b(h)592 b(that)g(its)1600 69557 y(complemen)-36 b(t)480 b(\(the)f(n)-36 b(um)g(b)36 b(ers)479 b Fl(\270)449 b Fm(n)19252 69756 y Fk(0)20258 69557 y Fo(that)480 b(are)h(not)f(in)g Fa(\256)p Fo(\))h(is)f(also)i (in\257nite,)491 b(its)481 b Fp(inverse)506 b(A)-33 b(r)-66 b(onson)1600 71162 y(tr)g(ansform)423 b Fo(is)i(the)f(sequence)h Fa(\257)414 b Fo(=)369 b Fl(f)p Fm(\257)74 b Fo(\()p Fm(n)p Fo(\))369 b(:)g Fm(n)h Fl(\270)f Fm(n)26803 71361 y Fk(0)27329 71162 y Fl(g)425 b Fo(suc)-36 b(h)424 b(that)g(the)g (Aronson)g(transform)h(of)g Fa(\257)471 b Fo(is)1600 72767 y Fa(\256)p Fo(.)p Black 25750 76088 a(11)p Black eop %%Page: 12 12 12 11 bop Black 0 TeXcolorgray Black Black Black 1600 2096 a Fn(Theorem)499 b(4.)p Black 651 w Fp(The)465 b(inverse)e(A)-33 b(r)-66 b(onson)464 b(tr)-66 b(ansform)464 b(exists)g(and)h(is)g (unique.)1600 5560 y Fn(Pro)42 b(of.)1301 b Fo(W)-108 b(e)450 b(establish)h(this)f(b)-36 b(y)451 b(giving)h(a)e(simple)h (algorithm)h(to)e(construct)g(the)g(in)-36 b(v)g(erse)451 b(trans-)1600 7165 y(form.)578 b(W)-108 b(e)431 b(illustrate)h(the)f (algorithm)g(in)h(T)-108 b(able)p 0 1 0 0 TeXcolorcmyk 431 w(I)36 b(I)p (#table.2) [[307 653 316 665] [1 1 1 [3 3]] [0 0 1]] pdfm Black 432 w(b)-36 b(y)431 b(applying)h(it)f(to)g(the)g(sequence)g(of)h (squares,)1600 8770 y Fa(\256)369 b Fo(=)g Fl(f)p Fm(n)5800 8288 y Fk(2)6695 8770 y Fo(:)h Fm(n)f Fl(\270)g Fo(0)p Fl(g)p Fo(.)3551 10375 y(F)-108 b(orm)544 b(a)g(table)h(with)f(four)h (ro)-36 b(ws.)911 b(In)544 b(the)f(\257rst)h(ro)-36 b(w)545 b(place)f(the)g(n)-36 b(um)g(b)36 b(ers)543 b Fm(n)558 b Fo(=)f Fm(n)45674 10574 y Fk(0)46200 10375 y Fo(,)572 b Fm(n)47909 10574 y Fk(0)48806 10375 y Fo(+)370 b(1,)1600 11980 y Fm(n)2376 12179 y Fk(0)3078 11980 y Fo(+)176 b(2)p Fm(;)221 b(:)g(:)g(:)s Fo(,)387 b(and)375 b(in)g(the)g(second)g (ro)-36 b(w)376 b(place)g(the)e(sequence)i Fm(\256)8 b Fo(\()p Fm(n)33358 12179 y Fk(0)33884 11980 y Fo(\),)387 b Fm(\256)8 b Fo(\()p Fm(n)37255 12179 y Fk(0)37958 11980 y Fo(+)176 b(1\))p Fm(;)221 b(:)g(:)g(:)s Fo(.)559 b(The)375 b(third)g(ro)-36 b(w)1600 13585 y(con)g(tains)488 b(what)h(w)-36 b(e)489 b(will)h(call)f(the)f(\\hot")h(n)-36 b(um)g(b)36 b(ers:)687 b(these)488 b(will)i(comprise)f(the)f(elemen)-36 b(ts)488 b(of)h(the)1600 15190 y(in)-36 b(v)g(erse)482 b(transform.)725 b(The)482 b(fourth)g(ro)-36 b(w)483 b(are)g(the)e(\\cold")j(n)-36 b(um)g(b)36 b(ers,)493 b(whic)-36 b(h)482 b(are)h(the)e(complemen)-36 b(t)1600 16795 y(of)434 b(the)f(hot)h(n)-36 b(um)g(b)36 b(ers.)3551 18400 y(The)332 b(third)e(and)i(fourth)f(ro)-36 b(ws)332 b(are)g(\257lled)f(in)h(as)g(follo)-36 b(ws.)546 b(If)333 b Fm(n)f Fo(is)g Fp(in)f Fo(\(resp.)543 b Fp(not)371 b(in)p Fo(\))331 b(the)g(sequence)1600 20005 y Fa(\256)p Fo(,)434 b(place)g Fm(\256)8 b Fo(\()p Fm(n)p Fo(\))434 b(in)g(the)f Fm(n)p Fo(-th)g(slot)h(of)g(the)f(hot)h(\(resp.)577 b(cold\))434 b(ro)-36 b(w.)3551 21611 y(T)-108 b(o)530 b(complete)f(the)g(table)g(w)-36 b(e)530 b(m)-36 b(ust)528 b(\257ll)i(in)f(the)f(empt)-36 b(y)529 b(slots.)866 b(Supp)36 b(ose)529 b(w)-36 b(e)529 b(are)h(at)f(column)1600 23216 y Fm(n)p Fo(,)590 b(where)559 b(w)-36 b(e)559 b(ha)-36 b(v)g(e)558 b(placed)h Fm(\256)8 b Fo(\()p Fm(n)p Fo(\))559 b(in)g(one)f(of)h(the)f(t)-36 b(w)g(o)559 b(slots.)954 b(Let)558 b Fm(l)37132 23415 y Fi(n)38317 23216 y Fo(b)36 b(e)558 b(the)g(largest)i(n)-36 b(um)g(b)36 b(er)1600 24821 y(men)-36 b(tioned)398 b(in)g(columns)g Fm(n)15176 25020 y Fk(0)15703 24821 y Fm(;)221 b(:)g(:)g(:)i(;)e(n)k Fl(\241)f Fo(1)399 b(in)f(the)g(hot)h(or)f(cold)h(ro)-36 b(ws.)567 b(Then)399 b(w)-36 b(e)399 b(place)f(the)h(n)-36 b(um)g(b)36 b(ers)1600 26426 y Fm(l)1987 26625 y Fi(n)2866 26426 y Fo(+)252 b(1)p Fm(;)221 b(:)g(:)g(:)j(;)d(\256)8 b Fo(\()p Fm(n)p Fo(\))254 b Fl(\241)e Fo(1)413 b(in)g(the)f(empt)-36 b(y)412 b(slot)h(in)g(column)f Fm(n)p Fo(,)417 b(with)c(the)f(single)h (exception)g(that)f(if)i Fm(n)f Fo(is)1600 28031 y(not)430 b(in)g(the)g(sequence)g(and)g Fm(l)15960 28230 y Fi(n)16955 28031 y Fo(=)369 b Fm(n)289 b Fl(\241)f Fo(1)431 b(then)e(w)-36 b(e)430 b(place)h Fm(n)f Fo(in)h(the)e(cold)i(slot)f(rather)g(than)g (the)f(hot)1600 29636 y(slot.)774 b(\(This)499 b(is)g(illustrated)g(b) -36 b(y)498 b(the)h(p)36 b(osition)499 b(of)g(2)g(in)g(the)f(fourth)g (ro)-36 b(w)500 b(of)f(T)-108 b(able)p 0 1 0 0 TeXcolorcmyk 499 w(I)36 b(I)p (#table.2) [[463 450 472 462] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.\))774 b(W)-108 b(e)499 b(lea)-36 b(v)g(e)1600 31241 y(it)549 b(to)h(the)f(reader)g(to)g(v)-36 b(erify)551 b(that)d(the)h(en)-36 b(tries)549 b(in)g(the)g(\\hot")h(ro)-36 b(w)549 b(form)h(the)f(in)-36 b(v)g(erse)549 b(Aronson)1600 32846 y(transform)434 b Fa(\257)45 b Fo(.)p 10730 32789 458 458 v Black 6870 34809 39061 45 v 6848 36414 45 1606 v 8842 35933 a Fm(n)p 11568 36414 V 2717 w Fo(1)p 13708 36414 V 1915 w(2)p 16698 36414 V 2374 w(3)p 19756 36414 V 3058 w(4)p 24115 36414 V 3709 w(5)p 28473 36414 V 3709 w(6)p 32832 36414 V 3708 w(7)p 37191 36414 V 3709 w(8)p 41549 36414 V 3709 w(9)p 45908 36414 V 6848 38019 V 8503 37538 a Fm(\256)9330 37737 y Fi(n)p 11568 38019 V 12335 37538 a Fo(1)p 13708 38019 V 1915 w(4)p 16698 38019 V 2374 w(9)p 19756 38019 V 2733 w(16)p 24115 38019 V 3059 w(25)p 28473 38019 V 3058 w(26)p 32832 38019 V 3059 w(49)p 37191 38019 V 3059 w(64)p 41549 38019 V 3058 w(81)p 45908 38019 V 6848 39624 V 7640 39143 a(\\hot")p 11568 39624 V 1516 w(1)p 13708 39624 V 1915 w(3)p 16698 39624 V 1724 w(5{8)p 19756 39624 V 2083 w(16)p 24115 39624 V 2083 w(17{24)p 28473 39624 V 1109 w(26{35)p 32832 39624 V 1109 w(37{48)p 37191 39624 V 1108 w(50{63)p 41549 39624 V 2084 w(81)p 45908 39624 V 6848 41230 V 7423 40748 a(\\cold")p 11568 41230 V 1108 w Fl(\241)p 13708 41230 V 1107 w Fo(2)p Fm(;)221 b Fo(4)p 16698 41230 V 1759 w(9)p 19756 41230 V 1758 w(10{15)p 24115 41230 V 2084 w(25)p 28473 41230 V 3058 w(36)p 32832 41230 V 3059 w(49)p 37191 41230 V 3059 w(64)p 41549 41230 V 2083 w(65{80)p 45908 41230 V 6870 41274 39061 45 v 1600 44133 a(T)-108 b(able)609 b(I)36 b(I:)p 0 TeXcolorgray Black 610 w(Computation)608 b(of)i(the)e(in)-36 b(v)g(erse)609 b(Aronson)f(transform)h(of)g(the)g (squares.)1104 b(The)609 b(\\hot")1600 45739 y(n)-36 b(um)g(b)36 b(ers)432 b(comprise)i(the)f(transform.)p Black 3551 48893 a(It)294 b(follo)-36 b(ws)296 b(from)e(Lemma)p 0 1 0 0 TeXcolorcmyk 294 w(1)p (#theorem.1) [[219 277 225 289] [1 1 1 [3 3]] [0 0 1]] pdfm Black 294 w(that)f Fa(\257)340 b Fo(con)-36 b(tains)293 b(the)h(mem)-36 b(b)36 b(ers)293 b(of)h Fa(\256)35996 48411 y Fk(\(2\))37253 48893 y Fo(,)323 b(but)292 b(in)i(general)g Fa(\257)415 b Fl(6)p Fo(=)369 b Fa(\256)49581 48411 y Fk(\(2\))50839 48893 y Fo(.)1600 50498 y(The)434 b(additional)g(terms)f (in)g Fa(\257)479 b Fo(mak)-36 b(e)434 b(it)g(p)36 b(ossible)434 b(to)g(reco)-36 b(v)g(er)434 b Fa(\256)g Fo(uniquely)g(from)g Fa(\257)45 b Fo(.)1600 53962 y Fn(Examples.)1302 b Fo(As)289 b(sho)-36 b(wn)289 b(in)g(T)-108 b(able)p 0 1 0 0 TeXcolorcmyk 289 w(I)36 b(I)p (#table.2) [[252 231 260 243] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)319 b(the)288 b(in)-36 b(v)g(erse)289 b(Aronson)g(transform)g (of)h(the)f(squares)g(\()p 0 1 0 0 TeXcolorcmyk(A10906)p [[496 231 534 243] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A010906) pdfm Black(\))1600 55567 y(is)15662 57173 y(1;)221 b(3;)g(5)p Fm(;)g Fo(6)p Fm(;)g Fo(7)p Fm(;)g Fo(8;)g(16;)g(17)p Fm(;)g(:)g(:)g(:)232 b(;)221 b Fo(24;)g(26)p Fm(;)g Fo(27)p Fm(;)g(:)g(:)g(:)449 b(:)1600 59497 y Fo(This)378 b(consists)g(of)h(a)f (n)-36 b(um)g(b)36 b(er)377 b(of)h(segmen)-36 b(ts)378 b(\(separated)g(here)f(b)-36 b(y)378 b(semicolons\).)560 b(F)-108 b(or)378 b Fm(k)414 b Fl(\270)369 b Fo(1)378 b(the)g Fm(k)50256 59015 y Fk(th)1600 61102 y Fo(segmen)-36 b(t)410 b(is)g Fl(f)p Fm(k)9275 60620 y Fk(2)9801 61102 y Fl(g)g Fo(if)h Fm(k)455 b Fo(is)410 b(a)g(square,)416 b(or)410 b Fl(f)p Fo(\()p Fm(k)292 b Fl(\241)247 b Fo(1\))26092 60620 y Fk(2)26865 61102 y Fo(+)f(1)p Fm(;)221 b(:)g(:)g(:)j(;)d(k) 32407 60620 y Fk(2)33181 61102 y Fl(\241)247 b Fo(1)p Fl(g)411 b Fo(if)f Fm(k)455 b Fo(is)411 b(not)e(a)i(square,)k(except) 1600 62707 y(that)433 b(the)g(second)g(segmen)-36 b(t)434 b(=)368 b Fl(f)p Fo(3)p Fl(g)p Fo(.)3551 64312 y(The)388 b(in)-36 b(v)g(erse)388 b(transform)g(of)h(the)e(primes)h(is)g(3,)398 b(5,)f(6,)h(11,)g(12,)g(17,)g(18,)g(20,)g(21,)g(22,)g Fm(:)221 b(:)g(:)390 b Fo(\()p 0 1 0 0 TeXcolorcmyk(A80759)p [[490 138 528 150] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A080759) pdfm Black(\))1600 65918 y(|)433 b(this)g(has)h(a)g(similar)g(decomp)36 b(osition)434 b(in)-36 b(to)434 b(segmen)-36 b(ts.)3551 67523 y(The)342 b(in)-36 b(v)g(erse)343 b(transform)f(of)h(the)f(lo)-36 b(w)g(er)342 b(Wytho\256)h(sequence)f(is)h(1,)361 b(4,)g(6,)g(7,)g(9,)g (10,)h(12,)f(14,)h(15,)f(17,)1600 69128 y Fm(:)221 b(:)g(:)324 b Fo(\()p 0 1 0 0 TeXcolorcmyk(A80746)p [[108 95 146 107] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A080746) pdfm Black(\).)543 b(This)323 b(consists)h(of)f(the)f(n)-36 b(um)g(b)36 b(ers)322 b Fl(b)p Fm(\301k)45 b Fl(c)69 b Fo(+)g Fm(k)114 b Fl(\241)69 b Fo(1)324 b(\()p Fm(k)414 b Fl(\270)369 b Fo(1\))323 b(and)f Fl(b)p Fo(2)p Fm(\301k)45 b Fl(c)69 b Fo(+)g Fm(k)114 b Fl(\241)69 b Fo(1)325 b(\()p Fm(k)413 b Fl(\270)370 b Fo(2\).)1600 70733 y(The)490 b(in)-36 b(v)g(erse)491 b(transform)g(of)g(our)f(original)i(sequence)e Fl(f)p Fm(a)p Fo(\()p Fm(n)p Fo(\))p Fl(g)h Fo(is)g(the)f(sequence)g (of)h(o)36 b(dd)490 b(n)-36 b(um)g(b)36 b(ers)1600 72338 y(\(whereas,)434 b(as)g(w)-36 b(e)434 b(sa)-36 b(w)434 b(in)f(Section)p 0 1 0 0 TeXcolorcmyk 434 w(2)p (#section.2) [[247 66 253 78] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)h Fm(a)21598 71856 y Fk(\(2\))23289 72338 y Fo(omits)g(3\).)p Black 25750 76088 a(12)p Black eop %%Page: 13 13 13 12 bop Black 0 TeXcolorgray Black Black 3551 2096 a Fo(In)619 b(general)h(\(b)36 b(ecause)619 b(of)h(the)f(ab)36 b(o)-36 b(v)g(e)620 b(algorithm\),)667 b(the)619 b(in)-36 b(v)g(erse)619 b(Aronson)g(transforms)h(are)1600 3701 y(easier)434 b(to)g(describ)36 b(e)433 b(than)g(the)g(direct)g (transforms.)1600 8138 y Fq(6.)1614 b(More)717 b(complicated)g (conditions)1600 11058 y Fo(Finally)-108 b(,)418 b(w)-36 b(e)413 b(ma)-36 b(y)414 b(mak)-36 b(e)414 b(the)e(condition)h(for)h Fm(n)f Fo(to)g(b)36 b(e)413 b(in)g(the)g(sequence)g(dep)36 b(end)411 b(on)i(the)g(v)-72 b(alues)413 b(of)1600 12664 y(sev)-36 b(eral)440 b(consecutiv)-36 b(e)440 b(terms)f Fm(a)p Fo(\()p Fm(n)p Fo(\),)i Fm(a)p Fo(\()p Fm(n)300 b Fo(+)f(1\))p Fm(;)221 b(:)g(:)g(:)i(;)e(a)p Fo(\()p Fm(n)300 b Fo(+)f Fm(\277)148 b Fo(\),)442 b(for)e(some)f(\257xed)h Fm(\277)148 b Fo(.)596 b(T)-108 b(o)440 b(pursue)e(this)1600 14269 y(further)395 b(w)-36 b(ould)396 b(tak)-36 b(e)397 b(us)f(in)-36 b(to)396 b(the)g(realm)g(of)h(one-dimensional)f(cellular) h(automata)g(\(cf.)566 b([)p 0 1 0 0 TeXcolorcmyk(9)p (#cite.Ila01) [[491 589 497 601] [1 1 1 [3 3]] [0 0 1]] pdfm Black(],)405 b([)p 0 1 0 0 TeXcolorcmyk(15)p (#cite.Wol02) [[510 589 522 601] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(]\),)1600 15874 y(and)433 b(w)-36 b(e)434 b(will)h(men)-36 b(tion)433 b(just)g(t)-36 b(w)g(o)434 b(examples.)3551 17479 y Fm(q)48 b Fo(\()p Fm(n)p Fo(\))376 b(is)h(the)g(smallest)g(p)36 b(ositiv)-36 b(e)378 b(in)-36 b(teger)377 b Fm(>)368 b(q)48 b Fo(\()p Fm(n)179 b Fl(\241)g Fo(1\))378 b(whic)-36 b(h)376 b(is)h(consisten)-36 b(t)377 b(with)g(the)f(condition)1600 19084 y(that)433 b(\\)p Fm(n)h Fo(is)g(in)g(the)f(sequence)g(if)i(and)e (only)h(if)g Fm(q)48 b Fo(\()p Fm(n)p Fo(\))433 b(is)h(o)36 b(dd)433 b(and)g Fm(q)48 b Fo(\()p Fm(n)295 b Fl(\241)g Fo(1\))434 b(is)g(ev)-36 b(en")434 b(\()p 0 1 0 0 TeXcolorcmyk(A79255)p [[471 545 509 557] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A079255) pdfm Black(\):)16579 22017 y(1)p Fm(;)221 b Fo(4)p Fm(;)g Fo(6)p Fm(;)g Fo(9)p Fm(;)g Fo(12)p Fm(;)g Fo(15)p Fm(;)g Fo(18)p Fm(;)g Fo(20)p Fm(;)g Fo(23)p Fm(;)g Fo(26)p Fm(;)g Fo(28)p Fm(;)g(:)g(:)g(:)1600 24951 y Fo(The)504 b(gaps)h(b)36 b(et)-36 b(w)g(een)504 b(successiv)-36 b(e)505 b(terms)f(are)h(alw)-36 b(a)g(ys)506 b(2)e(or)h(3.)791 b(Changing)505 b(the)f(condition)g(to)h(\\)p Fm(:)221 b(:)g(:)1600 26556 y Fo(b)36 b(oth)433 b Fm(q)48 b Fo(\()p Fm(n)p Fo(\))433 b(and)g Fm(q)48 b Fo(\()p Fm(n)295 b Fo(+)g(1\))433 b(are)h(o)36 b(dd")434 b(giv)-36 b(es)p 0 1 0 0 TeXcolorcmyk 434 w(A79259)p [[285 478 323 490] [1 1 1 [3 3]] [0 0 1]] 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(http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A001462) pdfm Black 0 1 0 0 TeXcolorcmyk 679 w(A079000)p 29662 35458 V [[339 400 383 412] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A079000) pdfm Black 0 1 0 0 TeXcolorcmyk 679 w(A079948)p 35216 35458 V [[389 400 433 412] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A079948) pdfm Black 0 1 0 0 TeXcolorcmyk 678 w(A080596)p 40769 35458 V [[439 400 483 412] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A080596) pdfm Black 0 1 0 0 TeXcolorcmyk 679 w(A079313)p 46323 35458 V [[489 400 533 412] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A079313) pdfm Black 0 1 0 0 TeXcolorcmyk 1600 36851 a(A079253)p 1600 37063 V [[86 385 130 397] [1 1 1 [3 3]] [0 0 1]] 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(http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A079258) pdfm Black 0 1 0 0 TeXcolorcmyk 291 w(A079254)p 32596 37063 V [[365 385 409 397] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A079254) pdfm Black 0 1 0 0 TeXcolorcmyk 291 w(A014132)p 37762 37063 V [[412 385 456 397] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A014132) pdfm Black 0 1 0 0 TeXcolorcmyk 292 w(A000217)p 42929 37063 V [[458 385 502 397] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A000217) pdfm Black 0 1 0 0 TeXcolorcmyk 291 w(A003605)p 48095 37063 V [[505 385 549 397] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A003605) pdfm Black 0 1 0 0 TeXcolorcmyk 1600 38456 a(A006166)p 1600 38668 V [[86 371 130 383] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A006166) pdfm Black 0 1 0 0 TeXcolorcmyk 291 w(A080637)p 6766 38668 V [[133 371 177 383] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A080637) pdfm Black 0 1 0 0 TeXcolorcmyk 291 w(A079882)p 11932 38668 V [[179 371 223 383] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A079882) pdfm Black 0 1 0 0 TeXcolorcmyk 291 w(A007378)p 17098 38668 V [[226 371 270 383] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A007378) pdfm Black 0 1 0 0 TeXcolorcmyk 291 w(A080780)p 22264 38668 V [[272 371 316 383] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A080780) pdfm Black 0 1 0 0 TeXcolorcmyk 291 w(A080588)p 27430 38668 V [[319 371 363 383] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A080588) pdfm Black 0 1 0 0 TeXcolorcmyk 291 w(A080591)p 32596 38668 V [[365 371 409 383] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A080591) pdfm Black 0 1 0 0 TeXcolorcmyk 291 w(A000201)p 37762 38668 V [[412 371 456 383] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A000201) pdfm Black 0 1 0 0 TeXcolorcmyk 292 w(A080760)p 42929 38668 V [[458 371 502 383] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A080760) pdfm Black 0 1 0 0 TeXcolorcmyk 291 w(A010906)p 48095 38668 V [[505 371 549 383] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A010906) pdfm Black 0 1 0 0 TeXcolorcmyk 1600 40061 a(A080759)p 1600 40273 V [[86 356 130 368] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A080759) pdfm Black 0 1 0 0 TeXcolorcmyk 436 w(A080746)p 6911 40273 V [[134 356 178 368] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A080746) pdfm Black 0 1 0 0 TeXcolorcmyk 435 w(A079255)p 12221 40273 V [[182 356 226 368] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A079255) pdfm Black 0 1 0 0 TeXcolorcmyk 436 w(A079259)p 17532 40273 V [[230 356 274 368] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A079259) pdfm Black 2 w(.\))p 1600 41766 49600 45 v 1600 44816 a(Receiv)-36 b(ed)400 b(Marc)-36 b(h)398 b(31,)407 b(2003;)412 b(revised)399 b(v)-36 b(ersion)400 b(receiv)-36 b(ed)399 b(July)g(2,)407 b(2003.)568 b(Published)398 b(in)h Fp(Journal)433 b(of)1600 46421 y(Inte)-66 b(ger)463 b(Se)-66 b(quenc)g(es)p Fo(,)431 b(July)j(4,)h(2003.)p 1600 48052 V 1600 50303 a(Return)e(to)p 0 1 0 0 TeXcolorcmyk 433 w(Journal)h(of)g(In)-36 b(teger)434 b(Sequences)f(home)g(page)p [[141 264 346 276] [1 1 1 [3 3]] [0 0 1]] (http://www.math.uwaterloo.ca/JIS/) pdfm Black(.)p Black 25750 76088 a(14)p Black eop %%Trailer end end