We introduce a new family of sequences {tk(n)}∞ n=-∞ for given positive integer k ≥ 3. We call these new sequences as generalized Alcuin's sequences because we get Alcuin's sequence which has several interesting properties when k = 3. Also, {tk(n)}∞ n=0 counts the number of partitions of n - k with parts being k, (k - 1),2(k - 1),3(k - 1),...,(k - 1) (k - 1). We find an explicit linear recurrence equation and the generating function for {tk(n)}∞ n=-∞. For the special case k = 4 and k = 5, we get a simpler formula for {tk(n)}∞ n=-∞ and investigate the period of {tk(n)}∞ n=-∞ modulo a fixed integer. Also, we get a formula for p5 (n) which is the number of partitions of n into exactly 5 parts.

Additional Metadata
Keywords Alcuin's sequence, Integer partition
Journal Electronic Journal of Combinatorics
Citation
Panario, D. (Daniel), Sahin, M. (Murat), & Wang, Q. (2012). Generalized Alcuin's sequence. Electronic Journal of Combinatorics, 19(4).