Sequences Associated With Balancing-Like Sequences

Abstract

The balancing-like sequences defined as 𝑥𝑛+1 = 𝐴𝑥𝑛 − 𝑥𝑛−1 with initializations 𝑥0 = 0, 𝑥1 = 1 (where 𝐴 > 2 is a natural number) are natural generalizations of the balancing sequence. It is an interesting idea to construct Lucas-balancing-like, cobalancing-like and Lucas-cobalancing-like sequences from balancing-like sequences and to see whether these sequences behave like Lucas-balancing, cobalancing and Lucas-cobalancing sequences respectively. Further, from each balancing-like sequence, it will be interesting extract two sequences (comparable to Pell and associated Pell sequences) such that the product of these sequences is equal to the corresponding balancing-like sequence. Generalized triangular numbers, with certain properties common with triangular numbers, can be constructed for each balancing-like sequence.