Generalized Alcuin's sequence (Q1953369)

Summary: 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.