Generalized Alcuin's Sequence

Abstract

We introduce a new family of sequences $\{t_k(n)\}_{n=-\infty}^{\infty}$ for given positive integer $k$. We call these new sequences asgeneralized Alcuin's sequences because we get Alcuin's sequence which has several interesting properties when $k=3$. Also, $\{t_k(n)\}_{n=0}^{\infty}$ counts the number of partitions of $n-k$ with parts being $k, \left(k-1\right), 2\left(k-1\right),$ $3\left(k-1\right)$, $\ldots, \left(k-1\right)\left(k-1\right)$. We find an explicit linear recurrence equation and the generating function for $\{t_k(n)\}_{n=-\infty}^{\infty}$. For the special case $k=4$ and $k=5$, we get a simpler formula for $\{t_k(n)\}_{n=-\infty}^{\infty}$ and investigate the period of $\{t_k(n)\}_{n=-\infty}^{\infty}$ modulo a fixed integer. Also, we get a formula for $p_{5}\left(n\right)$ which is the number of partitions of $n$ into exactly $5$ parts.