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

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

Offset

0,8

Comments

This sequence is motivated by the identity:

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

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).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..5000

Formula

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

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

Example

G.f.: A(x) = 1 + x^2 + x^4 + x^5 + x^6 + 2*x^7 + x^8 + 2*x^9 + 3*x^10 +...
where the g.f. may be expressed by the product:
A(x) = 1/((1-x^2)*(1-x^5)*(1-x^7)*(1-x^10)*(1-x^13)*...)
in which the exponents of x are the upper Wythoff numbers (A001950):
[2,5,7,10,13,15,18,20,23,26,28,31,34,36,39,41,44,47,49,52,54,57,60,...].
a(12) counts these partitions:  [10,2], [7,5], [5,5,2], [2,2,2,2,2,2. Clark Kimberling, Mar 09 2014

Mathematica

t = Table[Floor[n+n*GoldenRatio], {n, 1, 200}]; p[n_] := IntegerPartitions[n, All, t]; Table[ p[n], {n, 0, 12}] (*shows partitions*)
a[n_] := Length@p@n; a /@ Range[0, 80]
(* Clark Kimberling, Mar 09 2014 *)

Prog

(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)}

Crossrefs

Cf. A192184, A001950, A000041.

Sequence in context: A172986 A364642 A029826 * A246833 A213624 A080845

Adjacent sequences: A192182 A192183 A192184 * A192186 A192187 A192188

Keyword

nonn

Author

Paul D. Hanna, Jun 25 2011

Status

approved