Some Variants of the Balancing Sequence

Abstract

Balancing and cobalancing numbers admit generalizations in multiple directions. Sequence balancing numbers,gap balancing numbers,balancing-like numbers etc.are examples of such generalizations. The definition of cobalancing and balancing numbers involves balancing sums of natural numbers up to certain number and beyond the next or next to next number up to a feasible limit. If these sums are not exactly equal but differ by just unity, then the numbers in the positions of balancing and cobalancing numbers are termed as almost balancing and almost cobalancing numbers. Almost balancing as well as almost cobalancing numbers are governed by pairs of generalized Pell’s equation which are suitable alteration of the Pell’s equations for balancing and cobalancing numbers respectively. Similar alterations in the system of Pell’s equations of the balancing-like sequences result in a family of generalized Pell’s equation pair and their solutions result in almost balancing-like sequences. Another generalization of the notion of balancing numbers is possible by evenly arranging numbers on a circle (instead of arranging on a line) and deleting two numbers corresponding to a chord so as to balance the sums of numbers on the two resulting arcs. This consideration leads to the definition of circular balancing numbers. An interesting thing about studying several variations in the balancing sequence is that such variations increase the possibility of their application in other areas of mathematical sciences. For example,some of the balancing-like sequences along with their associated Lucas-balancing-like sequences are very closely associated with a statistical Diophantine problem.If the standard deviation 𝜎 of 𝑁 consecutive natural numbers is an integer then 𝜎 is twice some term of a balancing-like sequence and 𝑁,the corresponding term of the associated Lucas-balancing-like sequence.Also,these variations have many important unanswered aspects that would trigger future researchers to work in this area.