A332565 -id:A332565 - cp's OEIS frontend

A332022 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, n and a(n) have no common term in their Zeckendorf representations.

0, 2, 1, 5, 7, 3, 8, 4, 6, 13, 14, 15, 18, 9, 10, 11, 21, 23, 12, 24, 22, 16, 20, 17, 19, 34, 35, 36, 37, 38, 39, 40, 41, 47, 25, 26, 27, 28, 29, 30, 31, 32, 55, 57, 56, 60, 62, 33, 58, 59, 61, 63, 64, 65, 66, 42, 44, 43, 48, 49, 45, 50, 46, 51, 52, 53, 54, 89

Offset: 0

Rémy Sigrist, Apr 23 2020

nonn

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

Apparently, {a(0), ..., a(k)} = {0, ..., k} for infinitely many integers k.

			The first terms, alongside the Zeckendorf representation in binary of n and of a(n), are:
  n   a(n)  z(n)   z(a(n))
  --  ----  -----  -------
   0     0      0        0
   1     2      1       10
   2     1     10        1
   3     5    100     1000
   4     7    101     1010
   5     3   1000      100
   6     8   1001    10000
   7     4   1010      101
   8     6  10000     1001
   9    13  10001   100000
  10    14  10010   100001

Rémy Sigrist, Table of n, a(n) for n = 0..8360
Rémy Sigrist, Scatterplot of (x, y) such that x and y have no common term in their Zeckendorf representations and 0 <= x, y <= 1218
Rémy Sigrist, PARI program for A332022
Index entries for sequences that are permutations of the natural numbers

Cf. A003714, A238757 (binary analog), A332565.

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A003714(n) AND A003714(a(n)) = 0 for any n >= 0 (where AND denotes the bitwise AND operator).

A334348 The terms in the Zeckendorf representation of T(n, k) correspond to the terms in common in the Zeckendorf representations of n and of k; square array T(n, k) read by antidiagonals, n, k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 1, 0, 3, 3, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1, 5, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 5, 5, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 5, 6, 5, 0

Offset: 0

Rémy Sigrist, Apr 24 2020

This array has connections with the bitwise AND operator (A004198).

			Square array begins:
  n\k|  0  1  2  3  4  5  6  7  8  9  10  11  12  13
  ---+----------------------------------------------
    0|  0  0  0  0  0  0  0  0  0  0   0   0   0   0
    1|  0  1  0  0  1  0  1  0  0  1   0   0   1   0
    2|  0  0  2  0  0  0  0  2  0  0   2   0   0   0
    3|  0  0  0  3  3  0  0  0  0  0   0   3   3   0
    4|  0  1  0  3  4  0  1  0  0  1   0   3   4   0
    5|  0  0  0  0  0  5  5  5  0  0   0   0   0   0
    6|  0  1  0  0  1  5  6  5  0  1   0   0   1   0
    7|  0  0  2  0  0  5  5  7  0  0   2   0   0   0
    8|  0  0  0  0  0  0  0  0  8  8   8   8   8   0
    9|  0  1  0  0  1  0  1  0  8  9   8   8   9   0
   10|  0  0  2  0  0  0  0  2  8  8  10   8   8   0
   11|  0  0  0  3  3  0  0  0  8  8   8  11  11   0
   12|  0  1  0  3  4  0  1  0  8  9   8  11  12   0
   13|  0  0  0  0  0  0  0  0  0  0   0   0   0  13

Rémy Sigrist, Table of n, a(n) for n = 0..11475 (antidiagonals 0..150)
Rémy Sigrist, Colored representation of (x, y) for 0 <= x, y <= 1000 (where the hue is function of T(x, y))
Rémy Sigrist, PARI program for A334348
Wikipedia, Zeckendorf's theorem
Index entries for sequences related to Zeckendorf expansion of n

Cf. A003714, A022290, A004198, A332022, A332565.

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T(n, k) = A022290(A003714(n) AND A003714(k)) (where AND denotes the bitwise AND operator, A004198).
T(n, 0) = 0.
T(n, n) = n.
T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).

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

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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A370630 Lexicographically earliest sequence of distinct positive integers such that the Zeckendorf expansions of two consecutive terms have exactly one common term.

Original entry on oeis.org

1, 4, 3, 11, 8, 9, 6, 5, 7, 2, 10, 12, 14, 13, 15, 16, 18, 17, 20, 23, 21, 22, 19, 25, 26, 24, 27, 29, 28, 30, 35, 33, 37, 32, 38, 34, 36, 31, 41, 42, 39, 43, 47, 40, 44, 48, 45, 49, 46, 52, 60, 53, 56, 51, 58, 50, 59, 55, 57, 54, 61, 63, 62, 64, 68, 65, 69

Offset: 1

Rémy Sigrist, May 01 2024

Conjecture: this sequence is a permutation of the positive integers.

			The first terms, alongside the Zeckendorf expansion in binary of a(n), are:
  n   a(n)  z(a(n))
  --  ----  -------
   1     1        1
   2     4      101
   3     3      100
   4    11    10100
   5     8    10000
   6     9    10001
   7     6     1001
   8     5     1000
   9     7     1010
  10     2       10
  11    10    10010
  12    12    10101

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

Cf. A000120, A003714, A226077, A332565, A370631.

PARI
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A000120(A003714(a(n)), A003714(a(n+1))) = 1.

A352725 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the binary expansions of a(n) and a(n+1) have no common runs of consecutive 1's.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 12, 9, 14, 10, 13, 11, 15, 16, 24, 17, 26, 19, 25, 18, 27, 20, 28, 21, 30, 22, 29, 23, 31, 32, 48, 33, 50, 35, 49, 34, 51, 36, 54, 37, 55, 38, 52, 39, 53, 40, 56, 41, 58, 43, 57, 42, 59, 44, 60, 45, 62, 46, 61, 47, 63, 64, 96, 65, 98

Offset: 0

Rémy Sigrist, Mar 30 2022

This sequence is a variant of A109812; here we consider runs of consecutive 1's, there individual 1's in binary expansions.

The binary expansions of two consecutive terms may share some 1's, but cannot have a common run of consecutive 1's (as given by A352724).

			The first terms, alongside the corresponding partitions into runs of 1's, are:
  n   a(n)  runs in a(n)
  --  ----  ------------
   0     0  []
   1     1  [1]
   2     2  [2]
   3     3  [3]
   4     4  [4]
   5     6  [6]
   6     5  [1, 4]
   7     7  [7]
   8     8  [8]
   9    12  [12]
  10     9  [1, 8]
  11    14  [14]
  12    10  [2, 8]
  13    13  [1, 12]
  14    11  [3, 8]
  15    15  [15]
  16    16  [16]

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Scatterplot of the first 32769 terms
Rémy Sigrist, PARI program

Cf. A109812, A332565, A352724.

PARI
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A332022 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, n and a(n) have no common term in their Zeckendorf representations.

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A334348 The terms in the Zeckendorf representation of T(n, k) correspond to the terms in common in the Zeckendorf representations of n and of k; square array T(n, k) read by antidiagonals, n, k >= 0.

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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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A370630 Lexicographically earliest sequence of distinct positive integers such that the Zeckendorf expansions of two consecutive terms have exactly one common term.

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A352725 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the binary expansions of a(n) and a(n+1) have no common runs of consecutive 1's.

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