cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Rémy Sigrist, May 09 2024

Keywords

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.

Examples

			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

Links

Crossrefs

See A332565 for a similar sequence.
Cf. A356771, A361989, A372654, A372656 (inverse).

Programs

  • PARI
    \\ See Links section.

A372657 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the Fibonacci numbers that appear in the Zeckendorf representation of n are not missing from the dual Zeckendorf representation of a(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 19, 16, 21, 18, 20, 22, 23, 24, 25, 28, 29, 31, 26, 27, 32, 30, 33, 34, 35, 36, 38, 40, 37, 42, 39, 46, 48, 47, 51, 53, 41, 43, 44, 45, 56, 49, 50, 52, 54, 55, 57, 58, 59, 62, 63, 65, 60, 61, 66, 64, 67
Offset: 0

Views

Author

Rémy Sigrist, May 09 2024

Keywords

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 with inverse A372658: for any v >= 0, the majority of Fibonacci numbers are not missing from the dual Zeckendorf representation of v, and provide opportunities for v to be chosen, and so v will eventually appear in the sequence.

Examples

			The first terms, alongside the Zeckendorf representation of n and the dual Zeckendorf representation of a(n), in binary, are:
  n   a(n)  z(n)    d(a(n))
  --  ----  ------  -------
   0     0       0        0
   1     1       1        1
   2     2      10       10
   3     3     100       11
   4     4     101      101
   5     5    1000      110
   6     6    1001      111
   7     7    1010     1010
   8     8   10000     1011
   9     9   10001     1101
  10    10   10010     1110
  11    11   10100     1111
  12    12   10101    10101

Links

Crossrefs

See A372659 for a similar sequence.
Cf. A356771, A361989, A372658 (inverse).

Programs

  • PARI
    \\ See Links section.

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.

Original entry on oeis.org

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

Views

Author

Rémy Sigrist, May 09 2024

Keywords

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

Crossrefs

See A332022 for a similar sequence.
Cf. A356771, A361989, A372655.

Programs

  • PARI
    \\ See Links section.
Showing 1-3 of 3 results.

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