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A192184
Number of partitions of n into lower Wythoff numbers (A000201).
2
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
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 25 2011
STATUS
approved