login
A192185
Number of partitions of n into upper Wythoff numbers (A001950).
2
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
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 25 2011
STATUS
approved