A007000 Number of partitions of n into Fibonacci parts (with 2 types of 1).

1, 2, 4, 7, 11, 17, 25, 35, 49, 66, 88, 115, 148, 189, 238, 297, 368, 451, 550, 665, 799, 956, 1136, 1344, 1583, 1855, 2167, 2520, 2920, 3373, 3882, 4455, 5097, 5814, 6617, 7509, 8502, 9604, 10823, 12173, 13662, 15302, 17110, 19093, 21271, 23657, 26266

Offset: 0

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn

Examples

			a(2)=4 because we have [2],[1',1'],[1',1],[1,1] (the two types of 1 are denoted 1 and 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)
• James Propp and N. J. A. Sloane, Email, March 1994.

Crossrefs

Cf. A003107.

Cf. A000045.

Programs

Haskell

import Data.MemoCombinators (memo2, integral)
a007000 n = a007000_list !! n
a007000_list = map (p' 1) [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

with(combinat): gf := 1/product((1-q^fibonacci(k)), k=1..20): s := series(gf, q, 200): for i from 0 to 199 do printf(`%d,`,coeff(s, q, i)) od: # James Sellers, Feb 08 2002

Mathematica

CoefficientList[ Series[ 1/Product[1 - x^Fibonacci[i], {i, 1, 15}], {x, 0, 50}], x]
nmax = 46; f = Table[Fibonacci[n], {n, nmax}];
Table[Length[IntegerPartitions[n, All, f]], {n, 0, nmax}] (* Robert Price, Aug 02 2020 *)

Formula

a(n) = 1/n*Sum_{k=1..n} (A005092(k)+1)*a(n-k), n > 1, a(0)=1. - Vladeta Jovovic, Aug 22 2002

G.f.: 1/Product_{j>=1} (1-x^fibonacci(j)). - Emeric Deutsch, Mar 05 2006

G.f.: Sum_{i>=0} x^Fibonacci(i) / Product_{j=1..i} (1 - x^Fibonacci(j)). - Ilya Gutkovskiy, May 07 2017

Extensions

More terms from James Sellers, Feb 08 2002