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

1, 1, 1, 2, 3, 3, 5, 6, 8, 11, 13, 16, 23, 26, 32, 41, 50, 60, 75, 88, 108, 130, 154, 183, 222, 260, 307, 363, 429, 500, 589, 685, 800, 934, 1083, 1250, 1458, 1678, 1933, 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

Offset

0,4

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 A192185 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)), where phi = (sqrt(5)+1)/2.

G.f.: Product_{n>=1} 1/(1 - x^A000201(n)), where A000201 is the lower Wythoff sequence.

Example

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 +...
where the g.f. may be expressed by the product:
A(x) = 1/((1-x^1)*(1-x^3)*(1-x^4)*(1-x^6)*(1-x^8)*...)
in which the exponents of x are the lower Wythoff numbers (A000201):
[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,...].
a(7) counts these partitions:  61, 43, 4111, 331, 31111, 1111111. Clark Kimberling, Mar 09 2014

Mathematica

t = Table[Floor[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, PWL=1/prod(m=1, ceil(n/phi), 1-x^floor(m*phi)+x*O(x^n))); polcoeff(PWL, n)}

Crossrefs

Cf. A192185, A000201, A000041.

Sequence in context: A321286 A061790 A107236 * A027586 A339405 A039860

Adjacent sequences: A192181 A192182 A192183 * A192185 A192186 A192187

Keyword

nonn

Author

Paul D. Hanna, Jun 25 2011

Status

approved