A241737 Number of partitions p of n such that (number of numbers in p of form 3k+1) < (number of numbers in p of form 3k+2).

0, 0, 1, 0, 1, 2, 1, 2, 6, 3, 8, 14, 11, 20, 35, 31, 51, 77, 75, 113, 166, 168, 241, 333, 351, 482, 651, 697, 935, 1223, 1339, 1745, 2251, 2486, 3190, 4030, 4499, 5675, 7101, 7960, 9930, 12244, 13821, 17011, 20817, 23532, 28737, 34795, 39466, 47727, 57427

Offset: 0

Author

Clark Kimberling, Apr 28 2014

Keywords

• nonn
• easy

Comments

Each number in p is counted once, regardless of its multiplicity.

Examples

			a(8) counts these 6 partitions:  8, 61, 53, 521, 332, 2222.
		

Crossrefs

Cf. A241738, A241739, A241740, A241743.

Programs

Mathematica

z = 40; f[n_] := f[n] = IntegerPartitions[n]; s[k_, p_] := Count[Mod[DeleteDuplicates[p], 3], k];
Table[Count[f[n], p_ /; s[1, p] < s[2, p]], {n, 0, z}]  (* A241737 *)
Table[Count[f[n], p_ /; s[1, p] == s[2, p]], {n, 0, z}] (* A241738 *)
Table[Count[f[n], p_ /; s[1, p] > s[2, p]], {n, 0, z}]  (* A241739 *)

Formula

a(n) + A241738(n) + A241839(n) = A000041(n) for n >= 0.