

A286929


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


3



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
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OFFSET

0,3


COMMENTS

Generalization of AdamsWatters'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



