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.
@article{10_37236_2796,
author = {Daniel Panario and Murat Sahin and Qiang Wang},
title = {Generalized {Alcuin's} sequence},
journal = {The electronic journal of combinatorics},
year = {2012},
volume = {19},
number = {4},
doi = {10.37236/2796},
zbl = {1283.11029},
url = {http://geodesic.mathdoc.fr/articles/10.37236/2796/}
}
TY - JOUR
AU - Daniel Panario
AU - Murat Sahin
AU - Qiang Wang
TI - Generalized Alcuin's sequence
JO - The electronic journal of combinatorics
PY - 2012
VL - 19
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/2796/
DO - 10.37236/2796
ID - 10_37236_2796
ER -
%0 Journal Article
%A Daniel Panario
%A Murat Sahin
%A Qiang Wang
%T Generalized Alcuin's sequence
%J The electronic journal of combinatorics
%D 2012
%V 19
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/2796/
%R 10.37236/2796
%F 10_37236_2796
Daniel Panario; Murat Sahin; Qiang Wang. Generalized Alcuin's sequence. The electronic journal of combinatorics, Tome 19 (2012) no. 4. doi: 10.37236/2796
