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 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 (list; graph; refs; listen; history; text; internal format)
 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 Paul D. Hanna, Table of n, a(n) for n = 0..5000 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 Cf. A192184, A001950, A000041. Sequence in context: A071283 A172986 A029826 * A246833 A213624 A080845 Adjacent sequences:  A192182 A192183 A192184 * A192186 A192187 A192188 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 25 2011 STATUS approved

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Last modified February 28 01:24 EST 2020. Contains 332319 sequences. (Running on oeis4.)