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A239003 Number of partitions of n into distinct Fibonacci numbers that are all greater than 2.

%I A239003 #24 Mar 15 2014 22:07:55
%S A239003 1,0,0,1,0,1,0,0,2,0,0,1,0,2,0,0,2,0,1,0,0,3,0,0,2,0,2,0,0,3,0,0,1,0,
%T A239003 3,0,0,3,0,2,0,0,4,0,0,2,0,3,0,0,3,0,1,0,0,4,0,0,3,0,3,0,0,5,0,0,2,0,
%U A239003 4,0,0,4,0,2,0,0,5,0,0,3,0,3,0,0,4,0,0
%N A239003 Number of partitions of n into distinct Fibonacci numbers that are all greater than 2.
%C A239003 a(n) > 0 if n+1 is a term of A003622; a(n) = 0 if n+1 is a term of A022342.
%H A239003 Alois P. Heinz, <a href="/A239003/b239003.txt">Table of n, a(n) for n = 0..10946</a>
%F A239003 G.f.: product(1 + x^F(j), j=4..infinity). - _Wolfdieter Lang_, Mar 15 2014
%e A239003 There is one partition for n=0, the empty partition.  All parts are distinct, which means that there are no two parts that are equal. So a(0)=1.
%p A239003 F:= combinat[fibonacci]:
%p A239003 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<4, 0,
%p A239003        b(n, i-1)+`if`(F(i)>n, 0, b(n-F(i), i-1))))
%p A239003     end:
%p A239003 a:= proc(n) local j; for j from ilog[(1+sqrt(5))/2](n+1)
%p A239003        while F(j+1)<=n do od; b(n, j)
%p A239003     end:
%p A239003 seq(a(n), n=0..100);  # _Alois P. Heinz_, Mar 15 2014
%t A239003 f = Table[Fibonacci[n], {n, 4, 75}];  b[n_] := SeriesCoefficient[Product[1 + x^f[[k]], {k, n}], {x, 0, n}]; u = Table[b[n], {n, 0, 60}]  (* A239003 *)
%t A239003 Flatten[Position[u, 0]]  (* A022342 *)
%Y A239003 Cf. A000201, A001950, A000045, A000119, A239002, A000009.
%K A239003 nonn,easy,look
%O A239003 0,9
%A A239003 _Clark Kimberling_, Mar 08 2014