A286745 Number of distinct partitions of n with parts differing by at least two with smallest part at least two and with an odd number of parts.

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 13, 15, 17, 20, 22, 25, 28, 31, 34, 39, 42, 47, 52, 58, 64, 72, 79, 89, 99, 111, 123, 139, 154, 173, 193, 216, 240, 269, 298, 333, 369, 410, 453, 503, 554, 613, 674, 743, 815, 897, 981, 1077, 1177, 1288, 1405, 1536, 1672, 1825, 1985, 2163, 2350, 2558, 2776, 3019, 3275, 3557, 3856, 4186, 4534, 4919

Offset

0,13

Links

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

Formula

a(n) ~ exp(2*Pi*sqrt(n/15)) / (4 * 3^(1/4) * sqrt(5*phi) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 10 2020

Example

a(12) = 2 because of the partitions of 12, 12 and 6+4+2 are the only two that satisfy all three conditions.

Maple

b:= proc(n, i, t) option remember; `if`(n=0, t,
      `if`(i>n, 0, b(n, i+1, t)+b(n-i, i+2, 1-t)))
    end:
a:= n-> b(n, 2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Nov 23 2017

Mathematica

Table[Length@ Select[ip@n, Min[-Differences@#] >= 2 && Min@# >= 2 && OddQ@Length@# &], {n, 20}]

Crossrefs

Cf. A286744, A003106, A224898.

Sequence in context: A088670 A091581 A014591 * A286316 A027198 A238215

Adjacent sequences: A286742 A286743 A286744 * A286746 A286747 A286748

Keyword

nonn

Author

George Beck, May 13 2017

Status

approved