A192185 - OEIS

%I #14 Oct 21 2024 09:02:28

%S 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 31,42,37,50,51,57,70,69,83,91,95,120,118,139,153,161,193,200,224,254,

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

%C This sequence is motivated by the identity:

%C Product_{n>=1} (1 - x^[n*phi])*(1 - x^[n*phi^2]) / (1 - x^n) = 1, where [.] denotes floor(.).

%C 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 Paul D. Hanna, <a href="/A192185/b192185.txt">Table of n, a(n) for n = 0..5000</a>

%F G.f.: Product_{n>=1} 1/(1 - x^floor(n*phi^2)), where phi = (sqrt(5)+1)/2.

%F G.f.: Product_{n>=1} 1/(1 - x^A001950(n)), where A001950 is the upper Wythoff sequence.

%e 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 where the g.f. may be expressed by the product:

%e A(x) = 1/((1-x^2)*(1-x^5)*(1-x^7)*(1-x^10)*(1-x^13)*...)

%e in which the exponents of x are the upper Wythoff numbers (A001950):

%e [2,5,7,10,13,15,18,20,23,26,28,31,34,36,39,41,44,47,49,52,54,57,60,...].

%e a(12) counts these partitions: [10,2], [7,5], [5,5,2], [2,2,2,2,2,2]. _Clark Kimberling_, Mar 09 2014

%t t = Table[Floor[n+n*GoldenRatio], {n, 1, 200}]; p[n_] := IntegerPartitions[n, All, 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 *)

%o (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 Cf. A192184, A001950, A000041.

%K nonn

%O 0,8

%A _Paul D. Hanna_, Jun 25 2011