A372654 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the dual Zeckendorf representations of n and a(n) have no common missing Fibonacci number.

0, 1, 3, 2, 5, 4, 6, 9, 10, 7, 8, 11, 15, 16, 17, 12, 13, 14, 19, 18, 25, 26, 27, 29, 28, 20, 21, 22, 24, 23, 31, 30, 32, 41, 42, 43, 45, 44, 47, 46, 48, 33, 34, 35, 37, 36, 39, 38, 40, 51, 52, 49, 50, 53, 67, 68, 69, 71, 70, 73, 72, 74, 77, 78, 75, 76, 79, 54

Offset: 0

Author

Rémy Sigrist, May 09 2024

Keywords

• nonn
• base

Comments

We consider that a Fibonacci number is missing from the dual Zeckendorf representation of a number if it does not appear in this representation and a larger Fibonacci number appears in it.

The dual Zeckendorf representation is also known as the lazy Fibonacci representation (see A356771 for further details).

This sequence is a self-inverse permutation of the nonnegative integers.

Examples

			The first terms, alongside the dual Zeckendorf representation in binary of n and of a(n), are:
  n   a(n)  z(n)   z(a(n))
  --  ----  -----  -------
   0     0      0        0
   1     1      1        1
   2     3     10       11
   3     2     11       10
   4     5    101      110
   5     4    110      101
   6     6    111      111
   7     9   1010     1101
   8    10   1011     1110
   9     7   1101     1010
  10     8   1110     1011
  11    11   1111     1111
  12    15  10101    11010
  13    16  10110    11011
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10944
• Rémy Sigrist, Scatterplot of the sequence for n = 0..28655
• Rémy Sigrist, Scatterplot of (x, y) such that the dual Zeckendorf representations of x and y have no common missing term and x, y <= 1595
• 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

See A332022 for a similar sequence.

Cf. A356771, A361989, A372655.

Programs

PARI

\\ See Links section.