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

Author

George Beck, May 13 2017

Keywords

• nonn

Examples

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

Links

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

Crossrefs

Cf. A286744, A003106, A224898.

Programs

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}]

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