On several kinds of sums of balancing numbers

Abstract

The balancing numbers $B_n$ ($n=0,1,\cdots$) are solutions of the binary recurrence $B_n=6B_{n-1}-B_{n-2}$ ($n\ge 2$) with $B_0=0$ and $B_1=1$. In this paper we show several relations about the sums of product of two balancing numbers of the type ${\sum }_{m=0}^n{B}_{km+r}{B}_{k(n-m)+r}$ ($k>r\ge 0$) and the alternating sum of reciprocal of balancing numbers $\lfloor {\left({\sum }_{k=n}^{\infty }\frac{1}{B_{lk}}\right)}^{-1}\rfloor$. Similar results are also obtained for Lucas-balancing numbers $C_n$ ($n=0,1,\cdots$), satisfying the binary recurrence $C_n=6C_{n-1}-C_{n-2}$ ($n\ge 2$) with $C_0=1$ and $C_1=3$. Some binomial sums involving these numbers are also explored.

Ars Combin. 153 (2020), 127--147

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