A350240 Number of representations of n as a sum of distinct Fibonacci numbers where 1 can be included twice.

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

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):

a(n) = A000121(n) for n in A001950.

a(n) = A000119(n) for n-1 in A001950.

a(n) = A000121(n) - A000119(n) for n+1 in A001950.

Links

Table of n, a(n) for n=0..82.

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

Cf. A000045, A000119, A000121, A001950, A003107, A007000, A239002.

Sequence in context: A372433 A181630 A112310 * A342739 A137734 A366066

Adjacent sequences: A350237 A350238 A350239 * A350241 A350242 A350243

Keyword

nonn

Author

Kung Yue Tong, Dec 21 2021

Status

approved