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

0,3

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.

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

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.

Example

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.

Prog

(PARI) See Links section.

Crossrefs

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

Sequence in context: A344874 A051517 A289359 * A053118 A175682 A326722

Adjacent sequences: A356768 A356769 A356770 * A356772 A356773 A356774

Keyword

nonn,base

Author

Rémy Sigrist, Aug 27 2022

Status

approved