cp's OEIS Frontend

A192185 Number of partitions of n into upper Wythoff numbers (A001950).

%I A192185 #14 Oct 21 2024 09:02:28
%S A192185 1,0,1,0,1,1,1,2,1,2,3,2,4,3,5,6,5,8,7,9,13,10,16,14,18,22,21,28,29,
%T A192185 31,42,37,50,51,57,70,69,83,91,95,120,118,139,153,161,193,200,224,254,
%U A192185 262,312,324,360,404,427,485,525,561,640,668,758,817,878,982,1046,1150,1265,1340,1499,1597,1745,1911,2036,2241,2420,2602,2866,3041,3332,3597,3864,4221,4518
%N A192185 Number of partitions of n into upper Wythoff numbers (A001950).
%C A192185 This sequence is motivated by the identity:
%C A192185 Product_{n>=1} (1 - x^[n*phi])*(1 - x^[n*phi^2]) / (1 - x^n) = 1, where [.] denotes floor(.).
%C A192185 Therefore, the product of the g.f. of this sequence with the g.f. of A192184 yields the g.f. of the partition numbers (A000041).
%H A192185 Paul D. Hanna, <a href="/A192185/b192185.txt">Table of n, a(n) for n = 0..5000</a>
%F A192185 G.f.: Product_{n>=1} 1/(1 - x^floor(n*phi^2)), where phi = (sqrt(5)+1)/2.
%F A192185 G.f.: Product_{n>=1} 1/(1 - x^A001950(n)), where A001950 is the upper Wythoff sequence.
%e A192185 G.f.: A(x) = 1 + x^2 + x^4 + x^5 + x^6 + 2*x^7 + x^8 + 2*x^9 + 3*x^10 +...
%e A192185 where the g.f. may be expressed by the product:
%e A192185 A(x) = 1/((1-x^2)*(1-x^5)*(1-x^7)*(1-x^10)*(1-x^13)*...)
%e A192185 in which the exponents of x are the upper Wythoff numbers (A001950):
%e A192185 [2,5,7,10,13,15,18,20,23,26,28,31,34,36,39,41,44,47,49,52,54,57,60,...].
%e A192185 a(12) counts these partitions: [10,2], [7,5], [5,5,2], [2,2,2,2,2,2]. _Clark Kimberling_, Mar 09 2014
%t A192185 t = Table[Floor[n+n*GoldenRatio], {n, 1, 200}]; p[n_] := IntegerPartitions[n, All, t]; Table[ p[n], {n, 0, 12}] (*shows partitions*)
%t A192185 a[n_] := Length@p@n; a /@ Range[0, 80]
%t A192185 (* _Clark Kimberling_, Mar 09 2014 *)
%o A192185 (PARI) {a(n)=local(phi=(sqrt(5)+1)/2,PWU=1/prod(m=1,ceil(n/phi),1-x^floor(m*phi^2)+x*O(x^n)));polcoeff(PWU,n)}
%Y A192185 Cf. A192184, A001950, A000041.
%K A192185 nonn
%O A192185 0,8
%A A192185 _Paul D. Hanna_, Jun 25 2011