A332022 -id:A332022 - cp's OEIS frontend

A332565 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, a(n) and a(n+1) have no common term in their Zeckendorf representations.

0, 1, 2, 3, 5, 4, 7, 8, 6, 10, 13, 9, 15, 11, 14, 21, 12, 18, 22, 16, 23, 17, 26, 34, 19, 24, 20, 25, 36, 27, 37, 28, 35, 29, 38, 31, 39, 30, 41, 32, 40, 55, 33, 47, 56, 42, 57, 43, 58, 44, 59, 49, 60, 45, 61, 50, 62, 46, 68, 89, 48, 63, 51, 65, 52, 64, 54, 66

Offset: 0

Rémy Sigrist, Apr 23 2020

nonn

This sequence is a permutation of the natural numbers.

			The first terms, alongside their Zeckendorf representation in binary, are:
  n   a(n)  bin(A003714(a(n)))
  --  ----  ------------------
   0     0                   0
   1     1                   1
   2     2                  10
   3     3                 100
   4     5                1000
   5     4                 101
   6     7                1010
   7     8               10000
   8     6                1001
   9    10               10010
  10    13              100000

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program for A332565
Index entries for sequences that are permutations of the natural numbers

Cf. A003714, A109812 (binary analog), A332022.

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A003714(a(n)) AND A003714(a(n+1)) = 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).

A352726 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the binary expansions of n and a(n) have no common runs of consecutive 1's.

Original entry on oeis.org

0, 2, 1, 4, 3, 6, 5, 8, 7, 12, 13, 14, 9, 10, 11, 16, 15, 24, 25, 26, 27, 28, 29, 30, 17, 18, 19, 20, 21, 22, 23, 32, 31, 48, 49, 50, 51, 54, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 33, 34, 35, 36, 38, 39, 37, 40, 41, 42, 43, 44, 45, 46, 47, 64, 63, 96, 97, 98

Offset: 0

Rémy Sigrist, Mar 30 2022

This sequence is a self-inverse permutation of the nonnegative integers.
This sequence has similarities with A238757; here we consider runs of consecutive 1's, there individual 1's in binary expansions.
The binary expansion of n and a(n) 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 n  runs in a(n)
  --  ----  ---------  ------------
   0     0  []         []
   1     2  [1]        [2]
   2     1  [2]        [1]
   3     4  [3]        [4]
   4     3  [4]        [3]
   5     6  [1, 4]     [6]
   6     5  [6]        [1, 4]
   7     8  [7]        [8]
   8     7  [8]        [7]
   9    12  [1, 8]     [12]
  10    13  [2, 8]     [1, 12]
  11    14  [3, 8]     [14]
  12     9  [12]       [1, 8]
  13    10  [1, 12]    [2, 8]
  14    11  [14]       [3, 8]
  15    16  [15]       [16]
  16    15  [16]       [15]

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Scatterplot of the first 32769 terms
Rémy Sigrist, Scatterplot of (x, y) such that x, y < 2^10 and the binary expansions of x and y have no common runs of consecutive 1's
Rémy Sigrist, PARI program
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A238757, A332022, A352724.

PARI
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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

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 self-inverse permutation of the nonnegative integers.

			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

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

See A332022 for a similar sequence.

Cf. A356771, A361989, A372655.

PARI
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A370629 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, the Zeckendorf expansions of n and a(n) have exactly one common term.

Original entry on oeis.org

1, 2, 3, 6, 5, 4, 10, 8, 11, 7, 9, 14, 13, 12, 16, 15, 18, 17, 22, 23, 21, 19, 20, 26, 28, 24, 29, 25, 27, 35, 36, 37, 40, 34, 30, 31, 32, 39, 38, 33, 42, 41, 47, 48, 49, 52, 43, 44, 45, 58, 56, 46, 59, 57, 55, 51, 54, 50, 53, 63, 65, 64, 60, 62, 61, 68, 70

Offset: 1

Rémy Sigrist, May 01 2024

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

Fixed points correspond to positive Fibonacci numbers.

			The first terms, alongside the Zeckendorf expansion in binary of n and of a(n), are:
  n   a(n)  z(n)    z(a(n))
  --  ----  ------  -------
   1     1       1        1
   2     2      10       10
   3     3     100      100
   4     6     101     1001
   5     5    1000     1000
   6     4    1001      101
   7    10    1010    10010
   8     8   10000    10000
   9    11   10001    10100
  10     7   10010     1010
  11     9   10100    10001
  12    14   10101   100001

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
Index entries for sequences that are permutations of the natural numbers

Cf. A000120, A003714, A238758, A332022.

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

cp's OEIS Frontend

A332565 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, a(n) and a(n+1) 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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A352726 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the binary expansions of n and a(n) have no common runs of consecutive 1's.

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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.

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A370629 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, the Zeckendorf expansions of n and a(n) have exactly one common term.

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PARI

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