|
%I #15 Apr 29 2016 05:24:49
%S 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,
%T 109,130,146,164,195,217,245,285,319,357,415,461,517,592,660,735,840,
%U 931,1038,1175,1304,1446,1634,1805,2002,2246,2482,2742,3070,3381,3734
%N Number of partitions of n into parts congruent to 1, 3, 6, 8 (mod 9).
%C "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).
%H S. Kanade and M. C. Russell, <a href="http://arxiv.org/abs/1411.5346">IdentityFinder and some new identities of Rogers-Ramanujan type</a>, Exp. Math. 24:4 (2015), pp. 419-423.
%e 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.
%e 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.
%t Table[Length@ Select[IntegerPartitions@ n, AllTrue[Mod[#, 9], MemberQ[{1, 3, 6, 8}, #] &] &], {n, 0, 50}] (* _Michael De Vlieger_, Apr 28 2016, Version 10 *)
%Y Cf. A000726: partitions of 3n into parts == {3,6} mod 9.
%K nonn
%O 0,4
%A _Matthew C. Russell_, Apr 28 2016
|
