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

0,9

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(8) = 2 because 6+2 and 5+3 are the only partitions of 8 that satisfy the 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, 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 *)

Crossrefs

Cf. A286745, A003106, A224898, A001622.

Sequence in context: A264595 A026830 A025155 * A111684 A025159 A373073

Adjacent sequences: A286741 A286742 A286743 * A286745 A286746 A286747

Keyword

nonn

Author

George Beck, May 13 2017

Status

approved