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
Keywords
Examples
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
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..5000
Programs
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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 *) -
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)}
Comments