t-balancing numbers, Pell numbers and square triangular numbers

Abstract: Let $t\geq 2$ be an integer. In this work we get all integer solutions of the Diophantine equation $8r^2+8tr+1=y^2$ in order to determine the general terms of all t-balancing numbers for which $2t^{2}-1$ is prime. Later we obtain some formulas for the sums of Pell, Pell-Lucas, balancing and Lucas-balancing numbers in terms of t-balancing numbers and also we deduce the general terms of all t-balancing numbers in terms of square triangular numbers.

Keywords: Pell equation, balancing number, t-balancing number, square triangular number

Classification (MSC2000): 05A19; 11B37, 11B39

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