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User: Kung Yue Tong

Kung Yue Tong has authored 1 sequences.

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

Kung Yue Tong, Dec 21 2021

nonn

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.

			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.

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

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

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

cp's OEIS Frontend

User: Kung Yue Tong

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

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