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 A000121 Number of representations of n as a sum of Fibonacci numbers (1 is allowed twice as a part). (Formerly M0249 N0088) 32
 1, 2, 2, 3, 3, 3, 4, 3, 4, 5, 4, 5, 4, 4, 6, 5, 6, 6, 5, 6, 4, 5, 7, 6, 8, 7, 6, 8, 6, 7, 8, 6, 7, 5, 5, 8, 7, 9, 9, 8, 10, 7, 8, 10, 8, 10, 8, 7, 10, 8, 9, 9, 7, 8, 5, 6, 9, 8, 11, 10, 9, 12, 9, 11, 13, 10, 12, 9, 8, 12, 10, 12, 12, 10, 12, 8, 9, 12, 10, 13, 11, 9, 12, 9, 10, 11, 8, 9, 6, 6, 10, 9 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of partitions into distinct Fibonacci parts (1 counted as two distinct Fibonacci numbers). Inverse Euler transform of sequence has generating function sum_{n>0} x^F(n)-x^{2F(n)} where F() is Fibonacci. REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 0..6765 D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, part 1, part 2, Fib. Quart., 4 (1966), 289-306 and 322. Scott V. Tezlaf, On ordinal dynamics and the multiplicity of transfinite cardinality, arXiv:1806.00331 [math.NT], 2018. See p. 45. FORMULA a(0) = 1; for n >= 1, a(n) = A000119(n) + A000119(n-1). - Peter Munn, Jan 19 2018 MAPLE with(combinat): p := product((1+x^fibonacci(i)), i=1..25): s := series(p, x, 1000): for k from 0 to 250 do printf(`%d, `, coeff(s, x, k)) od: MATHEMATICA imax = 20; p = Product[1+x^Fibonacci[i], {i, 1, imax}]; CoefficientList[p, x][[1 ;; Fibonacci[imax]+1]] (* Jean-François Alcover, Feb 04 2016, adapted from Maple *) PROG (PARI) a(n)=local(A, m, f); if(n<0, 0, A=1+x*O(x^n); m=1; while((f=fibonacci(m))<=n, A*=1+x^f; m++); polcoeff(A, n)) CROSSREFS Cf. A000119. Least inverse is A083853. Sequence in context: A264982 A134674 A253894 * A230022 A049846 A086712 Adjacent sequences:  A000118 A000119 A000120 * A000122 A000123 A000124 KEYWORD nonn AUTHOR EXTENSIONS More terms from James A. Sellers, Jun 18 2000 STATUS approved

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Last modified October 19 09:28 EDT 2018. Contains 316339 sequences. (Running on oeis4.)