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A192184 Number of partitions of n into lower Wythoff numbers (A000201).

%I A192184 #18 Dec 02 2017 08:03:29
%S A192184 1,1,1,2,3,3,5,6,8,11,13,16,23,26,32,41,50,60,75,88,108,130,154,183,
%T A192184 222,260,307,363,429,500,589,685,800,934,1083,1250,1458,1678,1933,
%U A192184 2231,2565,2940,3381,3859,4418,5050,5753,6547,7464,8470,9617,10904,12352,13968,15801,17827,20115,22675,25531,28702,32288,36242,40664,45597,51079,57157
%N A192184 Number of partitions of n into lower Wythoff numbers (A000201).
%C A192184 This sequence is motivated by the identity:
%C A192184 Product_{n>=1} (1 - x^[n*phi])*(1 - x^[n*phi^2]) / (1 - x^n) = 1, where [.] denotes floor(.).
%C A192184 Therefore, the product of the g.f. of this sequence with the g.f. of A192185 yields the g.f. of the partition numbers (A000041).
%H A192184 Paul D. Hanna, <a href="/A192184/b192184.txt">Table of n, a(n) for n = 0..5000</a>
%F A192184 G.f.: Product_{n>=1} 1/(1 - x^floor(n*phi)), where phi = (sqrt(5)+1)/2.
%F A192184 G.f.: Product_{n>=1} 1/(1 - x^A000201(n)), where A000201 is the lower Wythoff sequence.
%e A192184 G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 5*x^6 + 6*x^7 + 8*x^8 +...
%e A192184 where the g.f. may be expressed by the product:
%e A192184 A(x) = 1/((1-x^1)*(1-x^3)*(1-x^4)*(1-x^6)*(1-x^8)*...)
%e A192184 in which the exponents of x are the lower Wythoff numbers (A000201):
%e A192184 [1,3,4,6,8,9,11,12,14,16,17,19,21,22,24,25,27,29,30,32,33,35,37,38,40,...].
%e A192184 a(7) counts these partitions:  61, 43, 4111, 331, 31111, 1111111. _Clark Kimberling_, Mar 09 2014
%t A192184 t = Table[Floor[n*GoldenRatio], {n, 1, 200}]; p[n_] := IntegerPartitions[n, All, t]; Table[ p[n], {n, 0, 12}] (*shows partitions*)
%t A192184 a[n_] := Length@p@n; a /@ Range[0, 80]
%t A192184 (* _Clark Kimberling_, Mar 09 2014 *)
%o A192184 (PARI) {a(n)=local(phi=(sqrt(5)+1)/2,PWL=1/prod(m=1,ceil(n/phi),1-x^floor(m*phi)+x*O(x^n)));polcoeff(PWL,n)}
%Y A192184 Cf. A192185, A000201, A000041.
%K A192184 nonn
%O A192184 0,4
%A A192184 _Paul D. Hanna_, Jun 25 2011