Properties of Sequences and Sums Associated with Balancing-like Sequences

Abstract

The balancing like sequences are natural generalizations of the balancing sequence with one exception that the balancing-like numbers do not arise out of any balancing problem. The balancing-like sequences also generalize the sequence of natural numbers and like the balancing numbers, they behave like natural numbers and in some identities. Also, in some identities, they behave like the trigonometric sine function. Certain recurrence sequences naturally arise in connection with the balancing sequence, namely, the Lucas-balancing sequence, the cobalancing sequence and the Lucas cobalancing sequence. Like the balancing numbers, the basic definition of cobalancing numbers involves a Diophantine equation comprising of natural numbers. However, the cobalancing numbers are nothing but the balancers–a type of numbers associated with the definition of balancing numbers. The Lucas-balancing numbers are related to the balancing numbers by means of a simple nonlinear relationship that leads to a Pell’s equation. The Lucas-cobalancing numbers appears with cobalancing numbers, exactly the way Lucas-balancing numbers appears with the balancing numbers. But, in some identities, the cobalancing, Lucas-balancing and Lucas-cobalancing numbers appear as linear combinations of the balancing numbers. Using similar linear relations, cobalancing-like and Lucas-cobalancing-like numbers are defined using a balancing-like sequence. The properties of these two new sequences are identical with that of the cobalancing and the Lucas-cobalancing msequences. The nth triangular number is defined as the sum of the first n natural numbers or the product of n and n + 1 divided by 2. The triangular numbers can be generalized to triangular-like numbers using a balancing-like sequence. The nth triangular-like number corresponding to a balancing-like sequence can be defined as the product of nth and (n+1)th term of the sequence divided by the second term. The nth triangular-like number, so defined, is not equal to the sum of first n terms of the corresponding balancing-like sequence, rather it is equal to the sum of first n terms of another balancing-like sequence. It is known that a natural number is a triangular number when eight times the number increased by 1, is a square. However, a natural number is a triangular-like number corresponding to a balancing-like sequence if two separate multiples of the numbers increased by 1, are perfect squares. A pronic number is the product of two consecutive natural numbers and a pronic-like number is defined as the product of any two consecutive terms of a balancing-like sequence. The balancing sequence is the only balancing-like sequence whose pronic-like numbers are both pronic and triangular and there are exactly two balancing-like sequences with triangular pronic-like numbers. Using the Fibonacci and Pell sequences, several balancing-like sequences can be constructed. Balancing-like sequences also appear in the products of some special Lucas sequences and their associated sequences