A286744 Number of distinct partitions of n with parts differing by at least two with smallest part at least two and with an even number of parts.

1, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 9, 9, 11, 12, 15, 16, 20, 22, 27, 30, 36, 40, 48, 53, 62, 69, 80, 88, 101, 111, 126, 138, 156, 170, 191, 208, 232, 253, 282, 306, 340, 370, 410, 446, 494, 537, 594, 647, 714, 778, 859, 935, 1031, 1124

Offset: 0

Author

George Beck, May 13 2017

Keywords

• nonn

Examples

			a(8) = 2 because 6+2 and 5+3 are the only partitions of 8 that satisfy the three conditions.
		

Links

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

Crossrefs

Cf. A286745, A003106, A224898, A001622.

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, 1):
seq(a(n), n=0..80);  # Alois P. Heinz, Nov 23 2017

Mathematica

Table[Length@
  Select[IntegerPartitions@n,
   Min[-Differences@#] >= 2 && Min@# >= 2 && EvenQ@Length@# &], {n,
  20}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, t,
     If[i > n, 0, b[n, i + 1, t] + b[n - i, i + 2, 1 - t]]];
a[n_] := b[n, 2, 1];
a /@ Range[0, 80] (* Jean-François Alcover, Jun 06 2021, after Alois P. Heinz *)

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