A372654 -id:A372654 - cp's OEIS frontend

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

Rémy Sigrist, May 09 2024

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.

			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

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

See A332565 for a similar sequence.

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

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

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