A225044 Number of partitions of n into non-triangular numbers, cf. A014132.

1, 0, 1, 0, 2, 1, 2, 2, 4, 4, 5, 6, 9, 10, 13, 14, 21, 22, 30, 32, 44, 48, 62, 69, 89, 100, 124, 141, 173, 198, 239, 274, 330, 377, 450, 514, 611, 697, 823, 939, 1104, 1258, 1470, 1676, 1950, 2220, 2572, 2927, 3381, 3841, 4420, 5019, 5759, 6529, 7470, 8460

Offset: 0

Author

Reinhard Zumkeller, Apr 25 2013

Keywords

• nonn

Examples

			a(10) = #{8+2, 5+5, 4+4+2, 4+2+2+2, 2+2+2+2+2} = 5;
a(11) = #{11, 9+2, 7+4, 7+2+2, 5+4+2, 5+2+2+2} = 6;
a(12) = #{12, 8+4, 8+2+2, 7+5, 5+5+2, 4+4+4, 4+4+2+2, 4+2+2+2+2, 6x2} = 9;
a(13) = #{13, 11+2, 9+4, 9+2+2, 8+5, 7+4+2, 7+2+2+2, 5+4+4, 5+4+2+2, 5+2+2+2+2} = 10.
		

Links

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

Crossrefs

Cf. A000217, A225045, A087153.

Column k=0 of A263234.

Programs

Haskell

a225044 = p a014132_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

Maple

b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(i>n or issqr(8*i+1), 0, b(n-i, i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 13 2015

Mathematica

t = Table[n (n + 1)/2, {n, 1, 200}] ; p[n_] := IntegerPartitions[n, All, Complement[Range@n, t]]; Table[p[n], {n, 0, 12}] (*shows partitions*)
a[n_] := Length@p@n; a /@ Range[0, 80]
(* Clark Kimberling, Mar 09 2014 *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n || IntegerQ @ Sqrt[8*i + 1], 0, b[n - i, i]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

Formula

G.f.: Product_{k>=1} (1 - x^(k*(k+1)/2))/(1 - x^k). - Ilya Gutkovskiy, Dec 30 2016

a(n) ~ sqrt(2) * exp(Pi*sqrt(2*n/3) - Zeta(3/2) * (3*n/2)^(1/4) - 3*Zeta(3/2)^2 / (16*Pi)) / sqrt(n). - Vaclav Kotesovec, Jan 01 2017