On the generalization neo balancing sequence and some applications

We investigate the generalization of sequence neo balancing numbers and their recurrence relations by extending the catalyst of certain sequences in balancing numbers to any integer k in the sequence neo balancing numbers. We derive the Diophantine equation for sequence neo balancing numbers in terms of k, which corresponds to the Diophantine equation for neo balancing numbers. We derive the Diophantine equation for the sequence neo balancing numbers and solve it via Pell's equation and Brahmagupta's identity. We examine the square root term in the derived Diophantine equation for the sequence neo balancing numbers by treating it as the generalized Pell's equation. Simultaneously, we consider the well-known Pell's equation. We integrate the generalized Pell's equation for the sequence neo balancing numbers to the well-known Pell's equation by using the Brahmagupta's identity. We obtain two solutions for both the generalized Pell's equation for the sequence neo balancing numbers and the well-known Pell's equation. The obtained solutions are sometimes analogous for some values of k. Then we investigate more precisely each case and substitute the solutions into the derived Diophantine equation for the sequence neo balancing numbers. Therefore, there are values of k that make both solutions analogous implying the recurrence relation can give all terms in the sequence neo balancing numbers. Simultaneously, there are values of k that make both solutions different implying that we need two recurrence relations generated by the two solutions to complete the sequence neo balancing numbers. Moreover, we establish a few theorems to explain why some values of k generate similar sequence of neo balancing numbers.