A372655 Lexicographically earliest sequence of distinct nonnegative integers such that the dual Zeckendorf representations of two consecutive terms have no common missing Fibonacci number.

0, 1, 2, 3, 4, 5, 6, 7, 9, 8, 10, 11, 12, 15, 14, 16, 13, 17, 18, 19, 20, 25, 21, 26, 22, 27, 23, 29, 24, 28, 30, 31, 32, 33, 41, 35, 42, 34, 43, 36, 45, 37, 44, 38, 47, 40, 46, 39, 48, 49, 51, 50, 52, 53, 54, 67, 55, 68, 56, 69, 57, 71, 58, 70, 59, 73, 61, 72

Offset

0,3

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 permutation of the nonnegative integers (as there as infinitely many numbers whose dual Zeckendorf representations have no missing Fibonacci number); see A372656 for the inverse.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10000

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

Example

The first terms, alongside their dual Zeckendorf representation in binary, are:
  n   a(n)  z(a(n))
  --  ----  -------
   0     0        0
   1     1        1
   2     2       10
   3     3       11
   4     4      101
   5     5      110
   6     6      111
   7     7     1010
   8     9     1101
   9     8     1011
  10    10     1110
  11    11     1111
  12    12    10101
  13    15    11010
  14    14    10111

Prog

(PARI) \\ See Links section.

Crossrefs

See A332565 for a similar sequence.

Cf. A356771, A361989, A372654, A372656 (inverse).

Sequence in context: A235489 A065306 A065307 * A356759 A372656 A375184

Adjacent sequences: A372652 A372653 A372654 * A372656 A372657 A372658

Keyword

nonn,base

Author

Rémy Sigrist, May 09 2024

Status

approved