Generalized Alcuin's Sequence

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.