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It is well known that, the problem of finding a sequence of real numbers an, n = 0, 1, 2, …, which is both geometric (an + 1 = kan; n = 0, 1, 2, …) and balancing (an + 1 = 6an − an − 1, a0 = 0, a1 = 1) admits an unique solution. In fact, the sequence is 1, λ1, λ12, …, λ1n, … where λ1 = 3 + √8 satisfies the balancing equation λ2 − 6λ + 1. In this paper, we pose an equivalent problem for a sequence of real, nonsingular matrices of order two and show that, this problem admits an infinity of solutions, that is there exist infinitely many such sequences.
- Balancing numbers
- Lucas-balancing numbers
- Balancing matrix
- Lucas-balancing matrix
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Cite this paper
Ray, P. K. (2014). Balancing sequences of matrices with application to algebra of balancing numbers. Notes on Number Theory and Discrete Mathematics, 20(1), 49-58.