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

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

Author

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

Keywords

• nonn
• easy

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

Examples

			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.
		

References

• 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.

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.

Programs

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

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 *)

PARI

f(x,y,z)=if(xCharles R Greathouse IV, Dec 14 2015

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 xReinhard 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

Extensions

More terms from Vladeta Jovovic, Jan 21 2002