A286316 Number of partitions of n with parts differing by at least two and with an odd number of parts.

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 13, 15, 17, 20, 22, 25, 28, 32, 35, 40, 44, 50, 56, 63, 70, 80, 89, 101, 113, 128, 143, 162, 181, 204, 228, 256, 285, 320, 355, 396, 439, 489, 540, 599, 660, 730, 803, 886, 972, 1070, 1172, 1287, 1408, 1544, 1686, 1846, 2014, 2202, 2400, 2621, 2854, 3114, 3389, 3693, 4016, 4374, 4753, 5172, 5617, 6107, 6628, 7201, 7810

Offset: 0

Author

George Beck, May 06 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A003114, A286041, A039924.

Programs

Mathematica

Table[Length@
  Select[IntegerPartitions@n, Min[-Differences@#] > 1 && OddQ@Length@# &], {n, 80}]

Formula

a(n) = A003114(n) - A286041(n).

a(n) ~ phi^(1/2) * exp(2*Pi*sqrt(n/15)) / (4 * 3^(1/4) * 5^(1/2) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 05 2020

Extensions

a(0)=0 prepended by Alois P. Heinz, May 07 2017