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

0,3

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

Links

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

Formula

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

Example

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

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 *)

Crossrefs

Cf. A126796.

Sequence in context: A240727 A123648 A244393 * A255525 A129632 A016028

Adjacent sequences: A286926 A286927 A286928 * A286930 A286931 A286932

Keyword

nonn

Author

Brian Hopkins, May 16 2017

Status

approved