A286929 Number of partitions of n such that each part is no more than 3 more than the sum of all smaller parts.

1, 1, 2, 3, 4, 6, 9, 13, 17, 25, 33, 46, 61, 84, 108, 147, 187, 248, 316, 414, 519, 675, 843, 1077, 1339, 1699, 2090, 2633, 3227, 4020, 4909, 6076, 7369, 9075, 10965, 13394, 16129, 19613, 23493, 28434, 33954, 40858, 48643, 58301, 69124, 82547, 97593, 116017, 136804, 162101, 190504

Offset: 0

Author

Brian Hopkins, May 16 2017

Keywords

• nonn

Comments

Generalization of Adams-Watters's criterion for complete partitions, that each part is no more than 1 more than the sum of all smaller parts.

Also the number of complete partitions of n+1 that contain more than one 1. - George Beck, Oct 01 2017

Examples

			For n = 8, a(8) = 17 counts all partitions of 8 except (8), (7,1), (6,2), (6,1,1), and (4,4).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A126796.

Programs

Mathematica

Table[Count[IntegerPartitions@ n, w_ /; And[Last@ w <= 3, NoneTrue[w - Rest@ PadRight[3 + Reverse@ Accumulate@ Reverse@ w, Length@ w + 1, Last@ w], # > 0 &]]], {n, 50}] (* Michael De Vlieger, May 16 2017, Version 10 *)

Formula

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n). - Vaclav Kotesovec, May 24 2018