A356759 Bit-reverse the odd part of the dual Zeckendorf representation of n: a(n) = A022290(A057889(A003754(n+1))).

0, 1, 2, 3, 4, 5, 6, 7, 9, 8, 10, 11, 12, 15, 17, 13, 16, 14, 18, 19, 20, 25, 22, 28, 30, 21, 26, 29, 23, 27, 24, 31, 32, 33, 41, 46, 36, 43, 38, 49, 51, 34, 42, 37, 47, 50, 35, 44, 48, 39, 45, 40, 52, 53, 54, 67, 59, 75, 80, 56, 70, 77, 62, 72, 64, 83, 85, 55

Offset: 0

Author

Rémy Sigrist, Aug 26 2022

Keywords

• nonn
• base
• look

Comments

This sequence is a self-inverse permutation of the nonnegative integers, similar to A345201 and A356331.

The dual Zeckendorf (or lazy Fibonacci) representation expresses uniquely a number n as a sum of distinct positive Fibonacci numbers; these distinct Fibonacci numbers can be encoded in binary, and the corresponding binary encoding, A003754(n+1), cannot have two consecutive nonleading 0's.

Examples

			For n = 49:
- the dual Zeckendorf representation of 49 is "1111010",
- reversing its odd part ("111101"), we obtain "1011110",
- so a(49) = 39.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10945
• Rémy Sigrist, PARI program
• Index entries for sequences related to Zeckendorf expansion of n
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A000045, A003714, A003754, A022290, A057889, A104326, A345201, A356331.

Programs

PARI

See Links section.

Formula

a(a(n)) = n.

a(n) < A000045(k) iff n < A000045(k).