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

%I A007000 M1045 #46 Jul 02 2025 16:01:55
%S A007000 1,2,4,7,11,17,25,35,49,66,88,115,148,189,238,297,368,451,550,665,799,
%T A007000 956,1136,1344,1583,1855,2167,2520,2920,3373,3882,4455,5097,5814,6617,
%U A007000 7509,8502,9604,10823,12173,13662,15302,17110,19093,21271,23657,26266
%N A007000 Number of partitions of n into Fibonacci parts (with 2 types of 1).
%D A007000 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007000 Reinhard Zumkeller, <a href="/A007000/b007000.txt">Table of n, a(n) for n = 0..10000</a> (first 1000 terms from T. D. Noe)
%H A007000 James Propp and N. J. A. Sloane, <a href="/A006997/a006997.pdf">Email, March 1994</a>.
%F A007000 a(n) = 1/n*Sum_{k=1..n} (A005092(k)+1)*a(n-k), n > 1, a(0)=1. - _Vladeta Jovovic_, Aug 22 2002
%F A007000 G.f.: 1/Product_{j>=1} (1-x^fibonacci(j)). - _Emeric Deutsch_, Mar 05 2006
%F A007000 G.f.: Sum_{i>=0} x^Fibonacci(i) / Product_{j=1..i} (1 - x^Fibonacci(j)). - _Ilya Gutkovskiy_, May 07 2017
%e A007000 a(2)=4 because we have [2],[1',1'],[1',1],[1,1] (the two types of 1 are denoted 1 and 1').
%p A007000 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
%t A007000 CoefficientList[ Series[ 1/Product[1 - x^Fibonacci[i], {i, 1, 15}], {x, 0, 50}], x]
%t A007000 nmax = 46; f = Table[Fibonacci[n], {n, nmax}];
%t A007000 Table[Length[IntegerPartitions[n, All, f]], {n, 0, nmax}] (* _Robert Price_, Aug 02 2020 *)
%o A007000 (Haskell)
%o A007000 import Data.MemoCombinators (memo2, integral)
%o A007000 a007000 n = a007000_list !! n
%o A007000 a007000_list = map (p' 1) [0..] where
%o A007000    p' = memo2 integral integral p
%o A007000    p _ 0 = 1
%o A007000    p k m | m < fib   = 0
%o A007000          | otherwise = p' k (m - fib) + p' (k + 1) m where fib = a000045 k
%o A007000 -- _Reinhard Zumkeller_, Dec 09 2015
%Y A007000 Cf. A003107.
%Y A007000 Cf. A000045.
%K A007000 nonn
%O A007000 0,2
%A A007000 _N. J. A. Sloane_, _Mira Bernstein_
%E A007000 More terms from _James Sellers_, Feb 08 2002