A356771 a(n) is the sum of the Fibonacci numbers in common in the Zeckendorf and dual Zeckendorf representations of n.

0, 1, 2, 0, 4, 0, 1, 7, 0, 1, 2, 3, 12, 0, 1, 2, 0, 4, 5, 6, 20, 0, 1, 2, 3, 4, 0, 1, 7, 8, 9, 10, 11, 33, 0, 1, 2, 0, 4, 5, 6, 7, 0, 1, 2, 3, 12, 13, 14, 15, 13, 17, 18, 19, 54, 0, 1, 2, 3, 4, 0, 1, 7, 8, 9, 10, 11, 12, 0, 1, 2, 0, 4, 5, 6, 20, 21, 22, 23, 24

Offset: 0

Author

Rémy Sigrist, Aug 27 2022

Keywords

• nonn
• base
• look

Comments

The Zeckendorf and dual Zeckendorf representations both express a number n as a sum of distinct positive Fibonacci numbers; these distinct Fibonacci numbers can be encoded in binary (see A022290 for the decoding function):

- in the Zeckendorf representation (or greedy Fibonacci representation):

- Fibonacci numbers are as big as possible (see A035517),

- and the corresponding binary encoding, A003714(n),

cannot have two consecutive 1's;

- in the dual Zeckendorf representation (or lazy Fibonacci representation):

- Fibonacci numbers are as small as possible (see A112309),

- and the corresponding binary encoding, A003754(n+1),

cannot have two consecutive nonleading 0's.

See A356326 for a similar sequence.

Examples

			For n = 28:
- using F(k) = A000045(k),
- the Zeckendorf representation of 28 is F(8) + F(5) + F(3),
- the dual Zeckendorf representation of 28 is F(7) + F(6) + F(5) + F(3),
- F(5) and F(3) appear in both representations,
- so a(28) = F(5) + F(3) = 7.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10946
• Rémy Sigrist, PARI program
• Index entries for sequences related to Zeckendorf expansion of n

Crossrefs

Cf. A000071, A003714, A003754, A022290, A035517, A112309, A331467, A356326.

Programs

PARI

\\ See Links section.

Formula

a(n) = A022290(A003714(n) AND A003754(n+1)) (where AND denotes the bitwise AND operator).

a(n) = 0 iff n belongs to A331467.

a(n) = n iff n belongs to A000071.