Sequence cobalacing and almost cobalancing numbers: A different approach.

The definition of balancing and cobalancing numbers are generalized to sequence balancing and cobalancing numbers as well as to almost balancing and cobalancing numbers. The sequence cobalancing numbers corresponding to the sequence {n + 1} for n > 0 asks for the existence of all pronic numbers which are 1 more than a triangular number. Similarly, the almost cobalancing numbers of first kind asks for all pronic numbers which are 1 less than a triangular number, whereas the second kind asks for the pronic numbers which are 1 more than a triangular number. Though the solutions of almost cobalancing numbers have been investigated using the theory of Pell's equation, the sequence cobalancing numbers for the sequence {n + 1} was not explored. In this paper, we study the problem of investigating the existence of all pronic and triangular numbers differing by 1 and we represent the solutions in terms of balancing and cobalancing numbers. However, our way of approaching to the solutions of Diophantine equations does not involve the theory of Pell's equation. [ABSTRACT FROM AUTHOR]