Cobalancing numbers and cobalancers

calling r ∈ Z+ the balancer corresponding to the balancing number n. The numbers 6, 35, and 204 are examples of balancing numbers with balancers 2, 14, and 84, respectively. Behera and Panda [1] also proved that a positive integer n is a balancing number if and only if n2 is a triangular number, that is, 8n2 + 1 is a perfect square. Though the definition of balancing numbers suggests that no balancing number should be less than 2, in [1], 1 is accepted as a balancing number being the positive square root of the square triangular number 1. In [4, 5], Subramaniam has explored some interesting properties of square triangular numbers. In a latter paper [6], he introduced the concept of almost square triangular numbers (triangular numbers that differ from a square by unity) and established links with the square triangular numbers. In this paper, we introduce cobalancing numbers and see that they are very closely associated with balancing numbers and also with triangular numbers which are products of two consecutive natural numbers. Observe that a number, which can be expressed as a product of two consecutive natural numbers, is almost equal to the arithmetic mean of squares of two consecutive natural numbers, that is, n(n+ 1) ≈ [n2 + (n+ 1)2]/2. In what follows, we introduce the cobalancing numbers in a way similar to the balancing numbers. By slightly modifying (1.1), we call n∈ Z+ a cobalancing number if 1 + 2 + ···+n= (n+ 1) + (n+ 2) + ···+ (n+ r) (1.2)