%I #25 May 20 2018 11:34:04
%S 1,1,2,3,4,6,8,10,14,18,23,29,37,47,59,74,92,114,141,173,213,261,318,
%T 387,470,569,687,827,994,1192,1426,1702,2028,2412,2863,3392,4012,4738,
%U 5585,6574,7726,9067,10624,12433,14528,16957,19763,23007,26749,31067,36034
%N Number of partitions of n into Fibonacci number of integer parts.
%C A003107 & this sequence are different sequences. A003107 gives the number of partitions in which each part of n is a Fibonacci number, this sequence gives the number of partitions in which the number of parts is a Fibonacci number. Both sequences share the same values for the first 9 values. For example A003107(4) = 4 because of the following 4 partitions of 5: (3,1), (2,2), (2,1,1), (1,1,1,1) whereas a(4) is also 4 but because of different set of partitions: (4), (3,1), (2,2), (2,1,1).
%H Alois P. Heinz, <a href="/A102848/b102848.txt">Table of n, a(n) for n = 0..5000</a>
%F G.f.: 1 + Sum_{n>=2} x^Fibonacci(n)/Product_{i=1..Fibonacci(n)} (1-x^i). - _Vladeta Jovovic_, Mar 02 2005
%e a(5) = 6 since out of 7 possible partitions of 5 into integer parts, only 6 include a Fibonacci number of parts: (5), (4,1), (3,2), (3,1,1), (2,2,1), (1,1,1,1,1). The 7th integer partitions of 5 (2,1,1,1) is not counted since it includes 4 integer parts and 4 is not a Fibonacci number.
%p b:= proc(n, i, t) option remember; `if`(n=0 or i=1,
%p `if`((h-> issqr(h+4) or issqr(h-4))(5*(t+n)^2), 1, 0),
%p b(n, i-1, t) + b(n-i, min(i, n-i), t+1))
%p end:
%p a:= n-> b(n$2, 0):
%p seq(a(n), n=0..80); # _Alois P. Heinz_, Jul 29 2017
%t b[n_, i_, t_] := b[n, i, t] = If[n == 0 || i == 1, If[IntegerQ @ Sqrt[# + 4] || IntegerQ @ Sqrt[# - 4]&[5*(t + n)^2], 1, 0], b[n, i - 1, t] + b[n - i, Min[i, n - i], t + 1]];
%t a[n_] := b[n, n, 0];
%t Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, May 20 2018, after _Alois P. Heinz_ *)
%Y Cf. A000040, A000045, A003107.
%K easy,nonn
%O 0,3
%A _Lior Manor_, Feb 28 2005
%E More terms from _Vladeta Jovovic_, Mar 02 2005
%E a(0)=1 prepended by _Alois P. Heinz_, Jul 29 2017