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

Author

Kung Yue Tong, Dec 21 2021

Keywords

• nonn

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.

Examples

			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.
		

Crossrefs

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

Programs

PARI

seq(n)=my(m=2); while(fibonacci(m)Andrew Howroyd, Dec 21 2021

Formula

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