

A350240


Number of representations of n as a sum of distinct Fibonacci numbers where 1 can be included twice.


0



1, 1, 2, 2, 2, 3, 2, 3, 3, 3, 4, 3, 3, 4, 3, 5, 4, 4, 5, 3, 4, 4, 4, 6, 5, 5, 6, 4, 6, 5, 5, 6, 4, 4, 5, 4, 7, 6, 6, 8, 5, 7, 6, 6, 8, 6, 6, 7, 5, 8, 6, 6, 7, 4, 5, 5, 5, 8, 7, 7, 9, 6, 9, 8, 8, 10, 7, 7, 8, 6, 10, 8, 8, 10, 6, 8, 7, 7, 10, 8, 8, 9, 6
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OFFSET

0,3


COMMENTS

A part of size 1 can be included twice in the partitions enumerated by this sequence, but there is only 1 way to include it once. The sequence A000119 only allows 1 to be included once and A000121 allows it to be included twice, but it in two different ways once.
Some connections with the upper Wythoff sequence (A001950):


LINKS



FORMULA

G.f.: (1 + x + x^2)*Product_{k>=3} (1 + x^Fibonacci(k)).  Andrew Howroyd, Dec 21 2021


EXAMPLE

The a(10)=4 partitions are: 8+2 = 8+1+1 = 5+3+1+1 = 5+3+2.
The a(11)=3 partitions are: 8+3 = 8+2+1 = 5+3+2+1.
The a(12)=3 partitions are: 8+3+1 = 8+2+1+1 = 5+3+2+1+1.


PROG

(PARI) seq(n)=my(m=2); while(fibonacci(m)<n, m++); Vec((1 + x + x^2 + O(x*x^n))*prod(k=3, m, 1 + x^fibonacci(k) + O(x*x^n))) \\ Andrew Howroyd, Dec 21 2021


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



