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%I #26 Mar 31 2017 09:35:17
%S 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,
%T 124,141,173,198,239,274,330,377,450,514,611,697,823,939,1104,1258,
%U 1470,1676,1950,2220,2572,2927,3381,3841,4420,5019,5759,6529,7470,8460
%N Number of partitions of n into non-triangular numbers, cf. A014132.
%H Alois P. Heinz, <a href="/A225044/b225044.txt">Table of n, a(n) for n = 0..10000</a>
%F G.f.: Product_{k>=1} (1 - x^(k*(k+1)/2))/(1 - x^k). - _Ilya Gutkovskiy_, Dec 30 2016
%F 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
%e a(10) = #{8+2, 5+5, 4+4+2, 4+2+2+2, 2+2+2+2+2} = 5;
%e a(11) = #{11, 9+2, 7+4, 7+2+2, 5+4+2, 5+2+2+2} = 6;
%e 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;
%e 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.
%p b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p b(n, i-1)+`if`(i>n or issqr(8*i+1), 0, b(n-i, i))))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Nov 13 2015
%t 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*)
%t a[n_] := Length@p@n; a /@ Range[0, 80]
%t (* _Clark Kimberling_, Mar 09 2014 *)
%t 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_ *)
%o (Haskell)
%o a225044 = p a014132_list where
%o p _ 0 = 1
%o p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
%Y Cf. A000217, A225045, A087153.
%Y Column k=0 of A263234.
%K nonn
%O 0,5
%A _Reinhard Zumkeller_, Apr 25 2013
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