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A000121 Number of representations of n as a sum of Fibonacci numbers (1 is allowed twice as a part).

%I A000121 M0249 N0088 #45 Jul 02 2025 16:01:53
%S A000121 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,
%T A000121 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,
%U A000121 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
%N A000121 Number of representations of n as a sum of Fibonacci numbers (1 is allowed twice as a part).
%C A000121 Number of partitions into distinct Fibonacci parts (1 counted as two distinct Fibonacci numbers).
%C A000121 Inverse Euler transform of sequence has generating function sum_{n>0} x^F(n)-x^{2F(n)} where F() is Fibonacci.
%D A000121 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000121 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000121 T. D. Noe, <a href="/A000121/b000121.txt">Table of n, a(n) for n = 0..6765</a>
%H A000121 Zai-Qiao Bai and Steven R. Finch, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/54-4/BaiFinch09122016.pdf">Fibonacci and Lucas Representations</a>, Fibonacci Quart. 54 (2016), no. 4, 319-326. See Table 1 p. 324.
%H A000121 D. A. Klarner, Representations of N as a sum of distinct elements from special sequences, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/4-4/klarner-a.pdf">part 1</a>, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/4-4/klarner-b.pdf">part 2</a>, Fib. Quart., 4 (1966), 289-306 and 322.
%H A000121 Scott V. Tezlaf, <a href="https://arxiv.org/abs/1806.00331">On ordinal dynamics and the multiplicity of transfinite cardinality</a>, arXiv:1806.00331 [math.NT], 2018. See p. 45.
%F A000121 a(0) = 1; for n >= 1, a(n) = A000119(n) + A000119(n-1). - _Peter Munn_, Jan 19 2018
%p A000121 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:
%t A000121 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 *)
%t A000121 nmax = 91; s=Total/@Subsets[Select[Table[Fibonacci[i], {i, nmax}], # <= nmax &]];
%t A000121 Table[Count[s, n], {n, 0, nmax}] (* _Robert Price_, Aug 17 2020 *)
%o A000121 (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))
%Y A000121 Cf. A000119, A274262, A356874.
%Y A000121 Least inverse is A083853.
%K A000121 nonn
%O A000121 0,2
%A A000121 _N. J. A. Sloane_
%E A000121 More terms from _James Sellers_, Jun 18 2000