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630 v 800 20267 a SDict begin H.S end 800 20267 a 800 20267 a SDict begin 14.5 H.A end 800 20267 a 800 20267 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.7) cvn H.B /DEST pdfmark end 800 20267 a Black 2084 x Ff(Theorem)499 b(7)p Black 651 w Fi(The)465 b(seventh)f(diagonal)g(c)-66 b(ontains)464 b(the)g Fh(n)p Fi(-dimensional)g(p)-66 b(artitions)464 b(of)g(4.)800 24807 y Ff(Pro)42 b(of.)654 b Fj(F)-108 b(or)408 b Fh(k)414 b Fj(=)369 b(7)409 b(,)k(w)-36 b(e)409 b(need)f(to)g(distribute)f Fh(n)244 b Fg(\000)g Fh(j)444 b Fj(=)368 b(6.)571 b(The)408 b(partitions)g(of)h(6)g(that)f (do)g(not)g(con)-36 b(tain)800 26412 y(a)566 b(1)h(are)f Fg(f)p Fj(6)p Fg(g)p Fj(,)600 b Fg(f)p Fj(4)p Fh(;)221 b Fj(2)p Fg(g)p Fj(,)601 b Fg(f)p Fj(3)p Fh(;)221 b Fj(3)p Fg(g)567 b Fj(and)f Fg(f)p Fj(2)p Fh(;)221 b Fj(2)p Fh(;)g Fj(2)p Fg(g)p Fj(.)978 b(There)566 b(are)g Fh(j)641 b Fj(comp)36 b(ositions)567 b(for)f(the)g(\014rst)f(partition,)800 28017 y Fh(j)75 b Fj(\()p Fh(j)275 b Fg(\000)200 b Fj(1\))387 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0 .5 0 TeXcolorrgb 8575 35209 a SDict begin H.R end 8575 35209 a 8575 36065 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.10) cvn H.B /ANN pdfmark end 8575 36065 a Black Fj(].)579 b(A008778)435 b(is)f(de\014ned)e(b)-36 b(y)18222 38483 y Fh(a)p Fj(\()p Fh(n)p Fj(\))369 b(=)f(\()p Fh(n)296 b Fj(+)e(1\)\()p Fh(n)27764 37935 y Fc(2)28586 38483 y Fj(+)h(8)p Fh(n)h Fj(+)e(6\))p Fh(=)p Fj(6)800 40902 y(and)490 b(coun)-36 b(ts)490 b(the)g Fh(n)p Fj(-dimensional)h(partitions)g(of)g(4)g(\(for)g (a)g(de\014nition)f(see)h([)p 0 .5 0 TeXcolorrgb 38977 40902 a SDict begin H.S end 38977 40902 a 0 .5 0 TeXcolorrgb Fj(1)p 0 .5 0 TeXcolorrgb 39627 40046 a SDict begin H.R end 39627 40046 a 39627 40902 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.1) cvn H.B /ANN pdfmark end 39627 40902 a Black Fj(],)506 b(p.)749 b(179\).)h(W)-108 b(e)491 b(need)f(to)800 42507 y(sho)-36 b(w)485 b(the)f(equiv)-72 b(alence)485 b(of)h(the)e(sequences)g 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Fj(2)800 65595 y(and)344 b(coun)-36 b(ts)343 b(the)h(p)36 b(en)-36 b(tagonal)345 b(p)-36 b(yramidal)344 b(n)-36 b(um)g(b)36 b(ers)343 b(\(for)i(a)f(de\014nition)g(see)g([)p 0 .5 0 TeXcolorrgb 38623 65595 a SDict begin H.S end 38623 65595 a 0 .5 0 TeXcolorrgb Fj(3)p 0 .5 0 TeXcolorrgb 39274 64739 a SDict begin H.R end 39274 64739 a 39274 65595 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.3) cvn H.B /ANN pdfmark end 39274 65595 a Black Fj(],)363 b(pp.)548 b(193-195\).)h(Clearly)-108 b(,)800 67200 y Fh(C)1731 67399 y Fl(j)2218 67200 y Fj(\()p Fh(j)369 b Fj(+)295 b(7)p Fh(;)6170 66916 y Fj(^)6170 67200 y(2)r(\))368 b(=)h Fh(a)p Fj(\()p Fh(j)75 b Fj(\).)p 52170 67200 630 630 v 800 67571 a SDict begin H.S end 800 67571 a 800 67571 a SDict begin 14.5 H.A end 800 67571 a 800 67571 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.9) cvn H.B /DEST pdfmark end 800 67571 a Black 26475 70521 a Fj(8)p Black eop end end %%Page: 9 9 TeXDict begin HPSdict begin 9 8 bop 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3906 y Fd(\023)42781 5779 y Fj(+)44087 3906 y Fd(\022)45084 4881 y Fh(j)45065 6690 y Fj(4)45716 3906 y Fd(\023)800 9141 y Fj(comp)36 b(ositions.)584 b Fh(C)10024 9340 y Fl(j)10511 9141 y Fj(\()p Fh(j)371 b Fj(+)296 b(8)p Fh(;)14466 8857 y Fj(^)14466 9141 y(2)q(\))435 b(app)36 b(ears)435 b(as)h(A008779)h(in)f(Sloane's)f(On-Line)f(Encyclop)36 b(edia)436 b(of)g(In)-36 b(teger)800 10747 y(Sequences)433 b([)p 0 .5 0 TeXcolorrgb 7274 10747 a SDict begin H.S end 7274 10747 a 0 .5 0 TeXcolorrgb Fj(10)p 0 .5 0 TeXcolorrgb 8575 9890 a SDict begin H.R end 8575 9890 a 8575 10747 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.10) cvn H.B /ANN pdfmark end 8575 10747 a Black Fj(].)579 b(A008779)435 b(is)f(de\014ned)e(b)-36 b(y)15374 13898 y Fh(a)p Fj(\()p Fh(n)p Fj(\))369 b(=)f(1)296 b(+)e(6)p Fh(n)i Fj(+)f(11)26175 12025 y Fd(\022)27154 13000 y Fh(n)27217 14810 y Fj(2)27930 12025 y Fd(\023)29203 13898 y Fj(+)g(7)31160 12025 y Fd(\022)32138 13000 y Fh(n)32201 14810 y 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Fj(+)301 b(1)442 b(in)h(the)e (expression)i(deriv)-36 b(ed)442 b(ab)36 b(o)-36 b(v)g(e)443 b(for)g Fh(C)34245 20459 y Fl(j)34732 20260 y Fj(\()p Fh(j)376 b Fj(+)300 b(8)p Fh(;)38696 19976 y Fj(^)38696 20260 y(2)r(\))442 b(and)g(simplifying)i(sho)-36 b(ws)800 21866 y(that)433 b Fh(C)4549 22065 y Fl(j)51 b Fc(+1)6238 21866 y Fj(\(\()p Fh(j)370 b Fj(+)294 b(1\))i(+)f(8)p Fh(;)13455 21581 y Fj(^)13455 21866 y(2)q(\))369 b(=)f Fh(a)p Fj(\()p Fh(j)75 b Fj(\),)433 b(i.e.,)i Fh(C)22851 22065 y Fl(j)23338 21866 y Fj(\()p Fh(j)369 b Fj(+)295 b(8)p Fh(;)27290 21581 y Fj(^)27290 21866 y(2)r(\))368 b(=)h Fh(a)p Fj(\()p Fh(j)h Fg(\000)295 b Fj(1\),)434 b(similar)g(to)g(the)f(case)h Fh(k)414 b Fj(=)369 b(7.)p 52170 21866 630 630 v 800 25076 a Ff(Remark.)845 b Fj(The)486 b Fh(n)p Fj(-dimensional)g(partitions)g(of)h(4)f(and)f(5)i(are)f(not)f (the)h(only)g(ones)g(con)-36 b(tained)486 b(in)g(the)800 26681 y(diagonals)427 b(of)f(T)-108 b(able)p 0 .5 0 TeXcolorrgb 11550 26681 a SDict begin H.S end 11550 26681 a 0 .5 0 TeXcolorrgb Fj(3)p 0 .5 0 TeXcolorrgb 12201 25825 a SDict begin H.R end 12201 25825 a 12201 26681 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (table.3) cvn H.B /ANN pdfmark end 12201 26681 a Black Fj(.)575 b(F)-108 b(urther)424 b(study)h(sho)-36 b(ws)426 b(that)g(the)f Fh(n)p Fj(-dimensional)h (partitions)f(of)i(2)f(and)f(3)h(also)800 28286 y(o)36 b(ccur)484 b(on)g(the)f(diagonals.)731 b(In)483 b([)p 0 .5 0 TeXcolorrgb 16911 28286 a SDict begin H.S end 16911 28286 a 0 .5 0 TeXcolorrgb Fj(1)p 0 .5 0 TeXcolorrgb 17562 27430 a SDict begin H.R end 17562 27430 a 17562 28286 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.1) cvn H.B /ANN pdfmark end 17562 28286 a Black Fj(],)497 b(form)-36 b(ulas)484 b(for)h(the)e Fh(n)p Fj(-dimensional)h(partitions)g(of)h Fh(k)529 b Fj(are)484 b(giv)-36 b(en)484 b(for)800 29891 y Fh(k)f Fg(\024)438 b Fj(6.)699 b(The)474 b Fh(n)p Fj(-dimensional)g 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Fg(\000)365 b Fj(1\).)883 b(A)535 b(pattern)f(emerges:)782 b(for)536 b(o)36 b(dd)535 b Fh(k)45 b Fj(,)561 b(the)800 36311 y Fh(k)1521 35829 y Fc(th)2983 36311 y Fj(diagonal)518 b(con)-36 b(tains)518 b(the)f Fh(n)p Fj(-dimensional)h(partitions)f(of) h(\()p Fh(k)398 b Fj(+)352 b(1\))p Fh(=)p Fj(2.)831 b(This)518 b(conjecture)f(w)-36 b(as)518 b(v)-36 b(ery)800 37916 y(exciting)458 b(b)36 b(ecause)457 b(no)g(generating)h(function)e (exists)i(for)g(this)f(family;)471 b(if)458 b(a)f(nice)g(connection)g (could)g(b)36 b(e)800 39522 y(established,)616 b(then)579 b(one)h(could)g(compute)f(the)g Fh(n)p Fj(-dimensional)h(partitions)f (in)h(a)g(simple)g(w)-36 b(a)g(y)-108 b(,)617 b(using)800 41127 y(Theorem)p 0 .5 0 TeXcolorrgb 6404 41127 a SDict begin H.S end 6404 41127 a 0 .5 0 TeXcolorrgb Fj(4)p 0 .5 0 TeXcolorrgb 7054 40271 a SDict begin H.R end 7054 40271 a 7054 41127 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.4) cvn H.B /ANN pdfmark end 7054 41127 a 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