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A067592 Number of partitions of n into Lucas parts (A000204(k) for k >= 1).

%I A067592 #26 Oct 08 2017 23:41:28
%S A067592 1,1,1,2,3,3,4,6,7,8,10,13,15,17,21,25,28,32,39,44,49,57,66,73,82,94,
%T A067592 105,116,130,147,162,178,199,221,241,265,295,322,350,385,423,458,498,
%U A067592 545,592,639,693,755,814,876,949,1026,1100,1183,1278,1371,1467,1576,1694,1809,1933,2072,2215,2359,2517,2691
%N A067592 Number of partitions of n into Lucas parts (A000204(k) for k >= 1).
%H A067592 Alois P. Heinz, <a href="/A067592/b067592.txt">Table of n, a(n) for n = 0..1000</a>
%F A067592 G.f.: 1/Product_{n>=1} (1 - q^A000204(n)). - _Joerg Arndt_, Mar 26 2014
%e A067592 a(7) counts these partitions: 7, 43, 4111, 331, 31111, 1111111. - _Clark Kimberling_, Mar 08 2014
%t A067592 p[n_] := IntegerPartitions[n, All, LucasL@Range@15]; Table[p[n], {n, 0, 12}] (* shows partitions *)
%t A067592 a[n_] := Length@p@n; a /@ Range[0,80] (* counts partitions, A067592 *)
%t A067592 (* _Clark Kimberling_, Mar 08 2014 *)
%t A067592 Table[SeriesCoefficient[gf = 1; k = 1; While[LucasL[k] <= n, gf = gf*(1 - x^LucasL[k]); k++]; gf = 1/gf, {x, 0, n}], {n, 0, 100}] (* _Vaclav Kotesovec_, Mar 26 2014, after _Joerg Arndt_ *)
%o A067592 (PARI) N=66; q='q+O('q^N);
%o A067592 L(n) = fibonacci(n+2) - fibonacci(n-2);
%o A067592 gf = 1; k=1; while( L(k) <= N, gf*=(1-q^L(k)); k+=1 ); gf = 1/gf;
%o A067592 Vec( gf ) /* _Joerg Arndt_, Mar 26 2014 */
%K A067592 easy,nonn
%O A067592 0,4
%A A067592 _Naohiro Nomoto_, Jan 31 2002