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Number of ordered factorizations of n into Fibonacci numbers > 1.
1

%I #7 May 11 2019 02:13:58

%S 1,1,1,1,1,2,0,2,1,2,0,3,1,0,2,3,0,3,0,3,1,0,0,6,1,2,1,0,0,6,0,4,0,1,

%T 0,6,0,0,2,6,0,2,0,0,3,0,0,11,0,3,0,3,0,4,1,0,0,0,0,12,0,0,2,6,2,0,0,

%U 2,0,0,0,13,0,0,3,0,0,6,0,11,1,0,0,3,0,0,0,0,1,12

%N Number of ordered factorizations of n into Fibonacci numbers > 1.

%F G.f. A(x) satisfies: A(x) = x + Sum_{k>=3} A(x^Fibonacci(k)).

%t terms = 90; A[_] = 0; Do[A[x_] = x + Sum[A[x^Fibonacci[k]], {k, 3, 25}] + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]

%t f[n_] := f[n] = SeriesCoefficient[Sum[x^Fibonacci[k], {k, 3, 25}], {x, 0, n}]; a[n_] := If[n == 1, n, Sum[If[d < n, f[n/d] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 90}]

%Y Cf. A000045, A010056, A065105 (positions of zeros), A065108 (positions of nonzero terms), A074206.

%K nonn

%O 1,6

%A _Ilya Gutkovskiy_, May 10 2019