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A225044 Number of partitions of n into non-triangular numbers, cf. A014132. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

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

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

EXAMPLE

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.

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-Fran├žois Alcover, Aug 29 2016, after Alois P. Heinz *)

PROG

(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

CROSSREFS

Cf. A000217, A225045, A087153.

Column k=0 of A263234.

Sequence in context: A163373 A117193 A026832 * A193691 A089408 A208888

Adjacent sequences:  A225041 A225042 A225043 * A225045 A225046 A225047

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Apr 25 2013

STATUS

approved

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Last modified October 23 14:06 EDT 2018. Contains 316528 sequences. (Running on oeis4.)