Trigonometric-type properties and the parity of balancing, Lucas-balancing, cobalancing and Lucas-cobalancing numbers

arXiv2004.05949MaRDI QIDQ6338604

Ngo van Dinh

Publication date: 13 April 2020

Abstract: Balancing numbers $n$ are originally defined as the solution of the Diophantine equation $1+2+cdots+(n-1)=(n+1)+cdots+(n+r)$, where $r$ is called the balancer corresponding to the balancing number $n$. By slightly modifying, $n$ is the cobalancing number with the cobalancer $r$ if $1+2+cdots+n=(n+1)+cdots+(n+r)$. Let $B_n$ denote the $n^{th}$ balancing number and $b_n$ denote the $n^{th}$ cobalancing number. Then $8B_n^2+1$ and $8b_n^2+8b_n+1$ are perfect squares. The $n^{th}$ Lucas-balancing number $C_n$ and the $n^{th}$ Lucas-cobalancing number $c_n$ are the positive roots of $8B_n^2+1$ and $8b_n^2+8b_n+1$, respectively. In this paper, we establish some trigonometric-type identities and some arithmetic properties concerning the parity of balancing, cobalancing, Lucas-balancing and Lucas-cobalancing numbers.

Mathematics Subject Classification ID

Recurrences (11B37) Special sequences and polynomials (11B83)

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