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 A003107 Number of partitions of n into Fibonacci parts (with a single type of 1). (Formerly M0556) 27
 1, 1, 2, 3, 4, 6, 8, 10, 14, 17, 22, 27, 33, 41, 49, 59, 71, 83, 99, 115, 134, 157, 180, 208, 239, 272, 312, 353, 400, 453, 509, 573, 642, 717, 803, 892, 993, 1102, 1219, 1350, 1489, 1640, 1808, 1983, 2178, 2386, 2609, 2854, 3113, 3393, 3697, 4017, 4367, 4737 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The partitions allow repeated items but the order of items is immaterial (1+2=2+1). - Ron Knott, Oct 22 2003 A098641(n) = a(A000045(n)). - Reinhard Zumkeller, Apr 24 2005 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe and Reinhard Zumkeller, Table of n, a(n) for n = 0..10000, first 1000 terms from T. D. Noe G. Almkvist, Partitions with Parts in a Finite Set and with Parts Outside a Finite Set, Exper. Math. vol 11 no 4 (2002) p 449-456. Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018. Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974. FORMULA a(n) = (1/n)*Sum_{k=1..n} A005092(k)*a(n-k), n > 1, a(0)=1. - Vladeta Jovovic, Jan 21 2002 G.f.: Product_{i>=2} 1/(1-x^fibonacci(i)). - Ron Knott, Oct 22 2003 a(n) = f(n,1,1) with f(x,y,z) = if x=2} x^Fibonacci(i) / Product_{j=2..i} (1 - x^Fibonacci(j)). - Ilya Gutkovskiy, May 07 2017 EXAMPLE a(4) = 4 since the 4 partitions of 4 using only Fibonacci numbers, repetitions allowed, are 1+1+1+1, 2+2, 2+1+1, 3+1. MAPLE F:= combinat[fibonacci]: b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0,        b(n, i-1)+`if`(F(i)>n, 0, b(n-F(i), i))))     end: a:= proc(n) local j; for j from ilog[(1+sqrt(5))/2](n+1)        while F(j+1)<=n do od; b(n, j)     end: seq(a(n), n=0..100);  # Alois P. Heinz, Jul 11 2013 MATHEMATICA CoefficientList[ Series[1/ Product[1 - x^Fibonacci[i], {i, 2, 21}], {x, 0, 53}], x] (* Robert G. Wilson v, Mar 28 2006 *) nmax = 53; s = Table[Fibonacci[n], {n, nmax}]; Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Jul 31 2020 *) F = Fibonacci; b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 2, 0,      b[n, i - 1] + If[F[i] > n, 0, b[n - F[i], i]]]]; a[n_] := Module[{j}, For[j = Floor@Log[(1+Sqrt[5])/2, n+1],      F[j + 1] <= n, j++]; b[n, j]]; a /@ Range[0, 100] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *) PROG (Haskell) import Data.MemoCombinators (memo2, integral) a003107 n = a003107_list !! n a003107_list = map (p' 2) [0..] where    p' = memo2 integral integral p    p _ 0 = 1    p k m | m < fib   = 0          | otherwise = p' k (m - fib) + p' (k + 1) m where fib = a000045 k -- Reinhard Zumkeller, Dec 09 2015 (PARI) f(x, y, z)=if(x

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Last modified June 29 18:25 EDT 2022. Contains 354913 sequences. (Running on oeis4.)