Neo balcobalancing numbers
Ahmet Tekcan
TL;DR
The paper introduces and develops the theory of neo balcobalancing numbers, defining $B_n^{neobc}$, $C_n^{neobc}$, $R_n^{neobc}$ and their Lucas variants, and shows that their general terms arise from Pell-type equations with $x^2-2y^2=-9$. It derives Binet-type formulas and recurrences for these sequences, and provides explicit parity-based mappings to the classical balancing and cobalancing sequences. It further connects the neo sequences to Pell and Pell-Lucas numbers, triangular and square-triangular numbers, Pythagorean triples, Cassini-type identities, and sums, yielding a unified framework that links these families and enables closed-form sums and interrelations.
Abstract
In this work, we defined neo balcobalancing numbers, neo Lucas-balcobalancing numbers, neo balcobalancers and neo Lucas-balcobalancers and derived the general terms of these numbers in terms of balancing numbers. Conversely we deduced the general terms of balancing, cobalancing, Lucas-balancing and Lucas-cobalancing numbers in terms of these numbers. We also deduced some relations on Binet formulas, recurrence relations, relationship with Pell, Pell-Lucas, triangular, square triangular numbers, Pythagorean triples and Cassini identities. We also formulate the sum of first $n$-terms of these numbers and obtained some formulas for the sums of Pell, Pell-Lucas, balancing and Lucas-cobalancing numbers in terms of these numbers.