A207033 Total number of parts >= 3 in all partitions of n.

0, 0, 1, 2, 4, 8, 13, 22, 35, 54, 80, 121, 172, 247, 347, 484, 661, 906, 1215, 1632, 2162, 2855, 3730, 4871, 6290, 8111, 10381, 13252, 16802, 21269, 26750, 33583, 41948, 52277, 64862, 80326, 99055, 121922, 149541, 183052, 223350, 272038, 330343, 400450, 484154

Offset: 1

Author

Omar E. Pol, Feb 18 2012

Keywords

• nonn

Examples

			a(4) = 2, because 2 parts have size >= 3 in all partitions of 4: [1,1,1,1], [1,1,2], [2,2], [1,3], [4].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Column 3 of A181187.

Cf. A006128, A096541, A206563.

Programs

Maple

b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, 0]
    elif i<1 then [0, 0]
    elif i>n then b(n, i-1)
    else f:= b(n, i-1); g:= b(n-i, i);
         [f[1]+g[1], f[2]+g[2] +`if`(i>2, g[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 19 2012

Mathematica

b[n_, i_] := b[n, i] = Module[{f, g}, Which[n == 0, {1, 0}, i < 1, {0, 0}, i > n, b[n, i - 1], True, f = b[n, i - 1]; g = b[n - i, i]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + If[i > 2, g[[1]], 0]}]];
a[n_] := b[n, n][[2]];
Array[a, 50] (* Jean-François Alcover, Nov 12 2020, after Alois P. Heinz *)

Formula

G.f.: Sum_{k>=1} x^(3*k)/(1 - x^k) / Product_{j>=1} (1 - x^j). - Ilya Gutkovskiy, Mar 05 2021

Extensions

More terms from Alois P. Heinz, Feb 18 2012