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

Author

Matthew C. Russell, Apr 28 2016

Keywords

• nonn

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).

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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 *)