A065108 -id:A065108 - cp's OEIS frontend

A261769 Number of distinct representations of n as a product of nonunit Fibonacci numbers (order doesn't matter).

1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 0, 1, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 1, 2, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 3, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 1, 0, 1

Offset: 1

Jeffrey Shallit, Nov 19 2015

nonn

			For n = 144 there are three representations: 144, 8*3*3*2, and 3*3*2*2*2*2, so a(144) = 3.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A065108.

A308062 Number of ordered factorizations of n into Fibonacci numbers > 1.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, May 10 2019

nonn

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

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]]
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}]

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

A286948 a(n) is the number of distinct products p of Fibonacci numbers such that Fibonacci(n) < p <= Fibonacci(n + 1).

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 5, 7, 10, 12, 17, 21, 29, 36, 48, 60, 78, 96, 124, 151, 190, 234, 290, 354, 436, 529, 648, 784, 952, 1141, 1382, 1651, 1984, 2367, 2831, 3359, 3999, 4733, 5608, 6614, 7816, 9178, 10802, 12667, 14850, 17356, 20297, 23653, 27579, 32062, 37277, 43235, 50139

Offset: 1

David A. Corneth, Jun 11 2017

nonn

The products of Fibonacci numbers are not all distinct. For example, 144 is a product of Fibonacci numbers in more than 1 way but 144 still counts as one product. 8 and 144 are the only Fibonacci numbers with this property (see A235383).
a(n+1)/a(n) seems to converge to about 1.1; up to n = 70, the value drops slightly. Maybe to sqrt(5)/2?
Fibonacci(n) ~= phi^n / sqrt(5) where phi = (1 + sqrt(5)) / 2. If m is a product of k Fibonacci numbers, m is of the form Fibonacci(n_1) *...* Fibonacci(n_k). To count the numbers just once, we restrict n_i for 1 <= i <= k.
Fibonacci(1) = Fibonacci(2) = 1 isn't counted, products with factor Fibonacci(6) = 8 aren't counted and products with the factor Fibonacci(12) = 144 aren't counted. I.e., n_i >= 3, n_i != 6 and n_i != 12.
We can write m uniquely as m = Product_{i=1..k} Fibonacci(n_i) ~= Product_{i=1..k} (phi^(n_i) / sqrt(5)) = phi^(Sum_{i=1..k} n_i) / sqrt(5)^k. To determine the number of such products up to f = Fibonacci(x) of k such Fibonacci factors, we can find an upper bound on Sum_{i=1..k} n_i of about (k*log(5)/2 + log(x)) / log(phi). This somewhat relates this sequence to the partitions.

			The products of Fibonacci numbers larger than Fibonacci(7) = 13 and smaller than or equal to Fibonacci(7 + 1) = 21 are the five numbers 15, 16, 18, 20 and 21. Therefore a(7) = 5.

Cf. A000041, A000045, A065108, A235383.

cp's OEIS Frontend

A261769 Number of distinct representations of n as a product of nonunit Fibonacci numbers (order doesn't matter).

Views

Author

Keywords

Examples

Links

Crossrefs

A308062 Number of ordered factorizations of n into Fibonacci numbers > 1.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A286948 a(n) is the number of distinct products p of Fibonacci numbers such that Fibonacci(n) < p <= Fibonacci(n + 1).

Views

Author

Keywords

Comments

Examples

Crossrefs