Periodicity of Balancing Numbers

Uniform Distribution of the Sequence of Balancing Numbers Modulo m

“…P r o o f. In [6], Panda et al shown that for any natural number n > 1, π(n) = n if and only if n = 2 k for some natural number k. It will be adequate to show that among the first 2 k elements of the sequence, we found at most one element from each residue class mod2 k . The basis step is clear by Theorem 2.6.…”

“…. Recently, Panda et al established the periodicity of balancing numbers modulo primes and studied the periods of sequence of balancing numbers modulo balancing, The Pell and the associated Pell numbers [6]. In addition, the authors also shown that the period of this sequence coincides with the modulus of congruence if there exists the modulus in any power of 2.…”

“…. It follows that,A (0, p; N ) = N λ + O(1),which ends the proof.For each natural number m, π(m) denotes the periodic length of the sequence of balancing numbers modulo m. In[6], Panda et al shown some divisibility properties concerning the periodicity of balancing numbers. Using those results, we have the following observations.Ç × ÖÚ Ø ÓÒ 2.8º We know that for any natural number m > 1, π(m) = m if and only if m = 2 k for some natural number k. Therefore, if the sequence of balancing numbers modulo m is uniform, then evidently m|π(m).…”

On Second Order Gap Balancing Numbers

“…The concept of gap balancing numbers was introduced by Panda and Rout [5] in connection with the Diophantine equation:…”

“…Motivated by higher order balancing numbers [4] and 2-gap balancing numbers [5], we introduce higher order -gap balancing number as follows.…”