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

Author

Paul D. Hanna, Jun 25 2011

Keywords

• nonn

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).

Examples

			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
		

Links

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

Crossrefs

Cf. A192185, A000201, A000041.

Programs

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 *)

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)}

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.