A272397 Number of partitions of n into parts congruent to 1, 3, 6, 8 (mod 9).

1, 1, 1, 2, 2, 2, 4, 4, 5, 7, 8, 9, 13, 14, 16, 21, 24, 27, 35, 39, 45, 55, 62, 70, 86, 96, 109, 130, 146, 164, 195, 217, 245, 285, 319, 357, 415, 461, 517, 592, 660, 735, 840, 931, 1038, 1175, 1304, 1446, 1634, 1805, 2002, 2246, 2482, 2742, 3070, 3381, 3734

Offset

0,4

Comments

"Sum side" conjecture: also equals number of partitions pi = (pi_1, pi_2, ...) of n (with pi_1 >= pi_2 >= ...) such that pi(i)-pi(i+2) >= 3 and, if pi(i) - pi(i+1) <= 1, then pi(i) + pi(i+1) is congruent to 0 (mod 3).

Links

Table of n, a(n) for n=0..56.

S. Kanade and M. C. Russell, IdentityFinder and some new identities of Rogers-Ramanujan type, Exp. Math. 24:4 (2015), pp. 419-423.

Example

For n=10, the a(10)=8 partitions are 10, 8+1+1, 6+3+1, 6+1+1+1, 3+3+3+1, 3+3+1+1+1+1. 3+1+1+1+1+1+1+1, and 1+1+1+1+1+1+1+1+1+1.
For the conjectured "sum side", the a(10)=8 partitions are 10, 9+1, 8+2, 7+3, 7+2+1, 6+4, 6+3+1, and 5+4+1.

Mathematica

Table[Length@ Select[IntegerPartitions@ n, AllTrue[Mod[#, 9], MemberQ[{1, 3, 6, 8}, #] &] &], {n, 0, 50}] (* Michael De Vlieger, Apr 28 2016, Version 10 *)

Crossrefs

Cf. A000726: partitions of 3n into parts == {3,6} mod 9.

Sequence in context: A029046 A035372 A035576 * A239729 A005859 A274143

Adjacent sequences: A272394 A272395 A272396 * A272398 A272399 A272400

Keyword

nonn

Author

Matthew C. Russell, Apr 28 2016

Status

approved