A003107 Number of partitions of n into Fibonacci parts (with a single type of 1). (Formerly M0556)

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

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

Batterman, Z. X., Jambhale, A., Miller, S. J., Narayanan, A. L., Sharma, K., Yang, A. K., & Yao, C. (2023). The Reversed Zeckendorf Game. arXiv preprint arXiv:2309.12748; Fib. Quart., Vol. 63:2 (2025), 215-239 (see page 228).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

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<y then 0^x else f(x-y,y,z)+f(x,y+z,y). - Reinhard Zumkeller, Nov 11 2009

G.f.: 1 + Sum_{i>=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<y, 0^x, f(x-y, y, z)+f(x, y+z, y))
a(n) = f(n, 1, 1) \\ Charles R Greathouse IV, Dec 14 2015

Crossrefs

Cf. A007000, A005092, A028290 (where the only Fibonacci numbers allowed are 1, 2, 3, 5 and 8).

Cf. A000045, A000119, A102848, A238998.

Row sums of A319394.

Sequence in context: A243225 A220851 A028290 * A217123 A014977 A008583

Adjacent sequences: A003104 A003105 A003106 * A003108 A003109 A003110

Keyword

nonn,easy

Author

N. J. A. Sloane, Herman P. Robinson

Extensions

More terms from Vladeta Jovovic, Jan 21 2002

Status

approved