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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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A133801 A181630 A112310 * A342739 A137734 A366066
KEYWORD
nonn
AUTHOR
Kung Yue Tong, Dec 21 2021
STATUS
approved

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Last modified February 28 07:05 EST 2024. Contains 370387 sequences. (Running on oeis4.)