A179046 Partitions into distinct parts with minimal difference 3 and minimal part >= 3.

1, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 23, 25, 28, 32, 35, 39, 44, 49, 54, 61, 67, 75, 83, 92, 101, 113, 123, 136, 150, 165, 180, 199, 217, 239, 261, 286, 312, 343, 373, 408, 445, 486, 528, 577, 626, 682, 741, 805, 873, 949, 1027, 1114

Offset: 0

Author

Joerg Arndt, Jan 04 2011

Keywords

• nonn

Examples

			a(13)=4 because there are 4 such partitions of 13: 3+10=4+9=5+8=13.
a(0)=1 because the condition is void for the empty list.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A003106 (min diff=2, min part=2), A000009 (min diff=1, min part=1).

Cf. A003114 (min diff=2), A025157 (min diff=3), A025158 (min diff=4), A025159 (min diff=5), A025160 (min diff=6), A025161 (min diff=7), A025162 (min diff=8).

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(n>i*(i+1)/2-3, 0, b(n, i-1)+
      `if`(i>n, 0, b(n-i, i-3))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 02 2014

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1,
  If[n > i(i+1)/2 - 3, 0, b[n, i - 1] +
  If[i > n, 0, b[n - i, i - 3]]]];
a[n_] := b[n, n];
a /@ Range[0, 80] (* Jean-François Alcover, Nov 20 2020, after Alois P. Heinz *)

PARI

N=66; x='x+O('x^N);
gf = sum(n=0,N, x^(3*n*(n+1)/2)/prod(k=1,n,1-x^k));
v = Vec(gf)
/* Joerg Arndt, Apr 07 2011 */

Sage

A179046 = lambda n: Partitions(n,max_slope=-3).filter(lambda x: not x or min(x) >= 3).cardinality() # D. S. McNeil, Jan 04 2011

Formula

G.f.: sum(n>=0, x^(3*n*(n+1)/2) / prod(k=1,n,1-x^k) ), this is a special case of the g.f. sum(n>=0, x^(D*n*(n+1)/2) / prod(k=1,n,1-x^k) ) for partitions into distinct parts with minimal difference D and minimal part >= D. - Joerg Arndt, Apr 07 2011

The g.f. for partitions into distinct parts with minimal difference D and no restriction on the minimal part is sum(n>=0, x^(D*n*(n+1)/2 - (D-1)*n) / prod(k=1..n, 1-x^k) ). - Joerg Arndt, Mar 31 2014