English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Balancing numbers $n$ are originally defined as the solution of theDiophantine equation $1+2+\cdots+(n-1)=(n+1)+\cdots+(n+r)$, where $r$ is calledthe 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 $n^{th}$ balancingnumber 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 sometrigonometric-type identities and some arithmetic properties concerning theparity of balancing, cobalancing, Lucas-balancing and Lucas-cobalancingnumbers.

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