A356771 -id:A356771 - cp's OEIS frontend

A361756 Irregular triangle T(n, k), n >= 0, k = 1..A361757(n), read by rows; the n-th row lists the numbers k such that the Fibonacci numbers that appear in the dual Zeckendorf representation of k also appear in that of n.

0, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 4, 0, 2, 5, 0, 1, 2, 3, 4, 5, 6, 0, 2, 7, 0, 1, 2, 3, 7, 8, 0, 1, 4, 9, 0, 2, 5, 7, 10, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 4, 12, 0, 2, 5, 13, 0, 1, 2, 3, 4, 5, 6, 12, 13, 14, 0, 2, 7, 15, 0, 1, 2, 3, 7, 8, 15, 16

Offset: 0

Rémy Sigrist, Mar 23 2023

In other words, the n-th row lists the numbers k such that A003754(1+n) AND A003754(1+k) = A003754(1+k) (where AND denotes the bitwise AND operator).

The dual Zeckendorf representation is also known as the lazy Fibonacci representation (see A356771 for further details).

			Triangle T(n, k) begins:
  n   n-th row
  --  -------------------------------------
   0  0
   1  0, 1
   2  0, 2
   3  0, 1, 2, 3
   4  0, 1, 4
   5  0, 2, 5
   6  0, 1, 2, 3, 4, 5, 6
   7  0, 2, 7
   8  0, 1, 2, 3, 7, 8
   9  0, 1, 4, 9
  10  0, 2, 5, 7, 10
  11  0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
  12  0, 1, 4, 12

Rémy Sigrist, Table of n, a(n) for n = 0..9956 (rows for n = 0..377 flattened)
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

See A361755 for a similar sequence.

Cf. A003754, A003842, A356771, A361757.

PARI
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T(n, 1) = 0.
T(n, 2) = A003842(n - 1) for any n > 0.
T(n, A361757(n)) = n.

A361989 a(n) is the sum of the Fibonacci numbers missing from the dual Zeckendorf representation of n; a(0) = 0, and for n > 0, a(n) = A022290(A035327(A003754(n+1))).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 0, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0

Offset: 0

Rémy Sigrist, Apr 02 2023

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 can also be seen as an irregular table T(n, k), n > 0, k = 1..A000045(n), where T(n, k) = A000045(n) - k.
a(n-1) for n>=1 is the starting position of the first occurrence of one of the longest words w in the Fibonacci word A003849 such that no length-n factor of w is repeated. The length of such words is 2n. (See links) - Gandhar Joshi, Mar 19 2024

			For n = 42:
- using F(k) = A000045(k),
- the dual Zeckendorf representation of 42 is F(8) + F(7) + F(5) + F(3) + F(2),
- the numbers F(6) and F(4) are missing,
- so a(42) = F(6) + F(4) = 8 + 3 = 11.
.
As an irregular triangle the sequence begins:
     0;
     0;
     1,  0;
     2,  1,  0;
     4,  3,  2, 1, 0;
     7,  6,  5, 4, 3, 2, 1, 0;
    12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0;
    ...

Gandhar Joshi, Walnut code and details
Index entries for sequences related to Zeckendorf expansion of n

Cf. A000045, A003754, A022290, A035327, A072649, A130312, A132665, A194029, A356771.

for (n = 1, 9, for (k = 1, f = fibonacci(n), print1 (f-k", ")))

a(n) = A000045(A072649(n)) - A194029(n) for n > 0.

a(n) = A130312(n) - A194029(n) for n > 0.

A361755 Irregular triangle T(n, k), n >= 0, k = 1..2^A007895(n), read by rows; the n-th row lists the numbers k such that the Fibonacci numbers that appear in the Zeckendorf representation of k also appear in that of n.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 1, 3, 4, 0, 5, 0, 1, 5, 6, 0, 2, 5, 7, 0, 8, 0, 1, 8, 9, 0, 2, 8, 10, 0, 3, 8, 11, 0, 1, 3, 4, 8, 9, 11, 12, 0, 13, 0, 1, 13, 14, 0, 2, 13, 15, 0, 3, 13, 16, 0, 1, 3, 4, 13, 14, 16, 17, 0, 5, 13, 18, 0, 1, 5, 6, 13, 14, 18, 19, 0, 2, 5, 7, 13, 15, 18, 20

Offset: 0

Rémy Sigrist, Mar 23 2023

In other words, the n-th row lists the numbers k such that A003714(n) AND A003714(k) = A003714(k) (where AND denotes the bitwise AND operator).

The Zeckendorf representation is also known as the greedy Fibonacci representation (see A356771 for further details).

			Triangle T(n, k) begins:
  n   n-th row
  --  ------------------------
   0  0
   1  0, 1
   2  0, 2
   3  0, 3
   4  0, 1, 3, 4
   5  0, 5
   6  0, 1, 5, 6
   7  0, 2, 5, 7
   8  0, 8
   9  0, 1, 8, 9
  10  0, 2, 8, 10
  11  0, 3, 8, 11
  12  0, 1, 3, 4, 8, 9, 11, 12

Rémy Sigrist, Table of n, a(n) for n = 0..10924 (rows for n = 0..610 flattened)
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

See A361756 for a similar sequence.

Cf. A003714, A007895, A139764, A295989, A356771.

PARI
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See Links section.
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T(n, 1) = 0.
T(n, 2) = A139764(n) for any n > 0.
T(n, 2^A007895(n)) = n.

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).

PARI
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\\ See Links section.
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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

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 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.

			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

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 A372659 for a similar sequence.

Cf. A356771, A361989, A372658 (inverse).

PARI
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A372659 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the Fibonacci numbers that appear in the Zeckendorf representation of n do not appear in the dual Zeckendorf representation of a(n).

Original entry on oeis.org

0, 2, 1, 3, 20, 4, 15, 12, 5, 7, 13, 8, 29, 6, 10, 21, 16, 36, 9, 19, 63, 11, 18, 17, 28, 33, 14, 26, 59, 22, 54, 56, 57, 101, 23, 34, 25, 27, 96, 46, 53, 88, 24, 44, 51, 42, 211, 38, 49, 93, 92, 180, 47, 91, 207, 30, 37, 64, 50, 62, 43, 60, 80, 31, 41, 85, 76

Offset: 0

Rémy Sigrist, May 09 2024

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 A372660.

			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     2       1        10
   2     1      10         1
   3     3     100        10
   4    20     101    101010
   5     4    1000       101
   6    15    1001    110110
   7    12    1010     10101
   8     5   10000       111
   9     7   10001      1110
  10    13   10010    101101
  11     8   10100      1011
  12    29   10101  10101010

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 A372657 for a similar sequence.

Cf. A356771, A372660 (inverse).

PARI
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\\ See Links section.
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A356965 a(n) is the sum of the tribonacci numbers in common in the greedy and lazy tribonacci representations of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 0, 8, 9, 10, 11, 12, 0, 1, 15, 16, 17, 18, 19, 13, 21, 22, 23, 0, 1, 2, 3, 28, 29, 30, 24, 32, 33, 34, 35, 36, 24, 25, 39, 40, 41, 42, 43, 0, 1, 2, 3, 4, 5, 6, 7, 52, 53, 54, 55, 56, 44, 45, 59, 60, 61, 62, 63, 57, 65, 66, 67, 44, 45, 46

Offset: 0

Rémy Sigrist, Sep 06 2022

This sequence is to tribonacci numbers (A000073) what A356771 is to Fibonacci numbers (A000045).

			For n = 58:
- with T = A000073,
- the greedy representation of 58 is: T(9) + T(7) + T(3),
- the lazy representation of 58 is: T(9) + T(6) + T(5) + T(4) + T(3),
- so a(59) = T(9) + T(3) = 45.

Rémy Sigrist, Table of n, a(n) for n = 0..10609
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

Cf. A000045, A000073, A003726, A003796, A356771, A356899, A356964, A356966.

PARI
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See Links section.
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a(n) = A356964(A003796(n+1) AND A003726(n+1)) (where AND denotes the bitwise AND operator).
a(n) <= n with equality iff n belongs to A356899.
a(n) = 0 iff n belongs to A356966.

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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\\ See Links section.
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A361789 A(n, k) is the sum of the distinct terms in the dual Zeckendorf representations of n or of k; square array A(n, k) read by antidiagonals, n, k >= 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 3, 3, 3, 4, 3, 2, 3, 4, 5, 4, 3, 3, 4, 5, 6, 6, 6, 3, 6, 6, 6, 7, 6, 5, 6, 6, 5, 6, 7, 8, 8, 6, 6, 4, 6, 6, 8, 8, 9, 8, 7, 6, 6, 6, 6, 7, 8, 9, 10, 9, 8, 8, 6, 5, 6, 8, 8, 9, 10, 11, 11, 11, 8, 11, 6, 6, 11, 8, 11, 11, 11, 12, 11, 10, 11, 11, 10, 6, 10, 11, 11, 10, 11, 12

Offset: 0

Rémy Sigrist, Mar 24 2023

The dual Zeckendorf representation corresponds to the lazy Fibonacci representation (see A356771 for further details).

			Array A(n, k) begins:
  n\k |  0   1   2   3   4   5   6   7   8   9  10  11  12  13
  ----+-------------------------------------------------------
    0 |  0   1   2   3   4   5   6   7   8   9  10  11  12  13
    1 |  1   1   3   3   4   6   6   8   8   9  11  11  12  14
    2 |  2   3   2   3   6   5   6   7   8  11  10  11  14  13
    3 |  3   3   3   3   6   6   6   8   8  11  11  11  14  14
    4 |  4   4   6   6   4   6   6  11  11   9  11  11  12  14
    5 |  5   6   5   6   6   5   6  10  11  11  10  11  14  13
    6 |  6   6   6   6   6   6   6  11  11  11  11  11  14  14
    7 |  7   8   7   8  11  10  11   7   8  11  10  11  19  18
    8 |  8   8   8   8  11  11  11   8   8  11  11  11  19  19
    9 |  9   9  11  11   9  11  11  11  11   9  11  11  17  19
   10 | 10  11  10  11  11  10  11  10  11  11  10  11  19  18
   11 | 11  11  11  11  11  11  11  11  11  11  11  11  19  19
   12 | 12  12  14  14  12  14  14  19  19  17  19  19  12  14
   13 | 13  14  13  14  14  13  14  18  19  19  18  19  14  13

Rémy Sigrist, Colored representation of the table for x, y < Fibonacci(16) - 1 (where the color is function of A(x, y))
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

Cf. A003754, A003986, A022290, A334348, A356771, A356969, A361756.

PARI
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See Links section.
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A(n, k) = A022290(A003754(n+1) OR A003754(k+1)) (where OR denotes the bitwise OR operator, A004198).
A(n, k) = A(k, n).
A(n, 0) = n.
A(n, n) = n.
A(A(m, n), k) = A(m, A(n, k)).
A(A(n, k), n) = A(n, k).
A(n, A361756(n, k)) = n.

cp's OEIS Frontend

A361756 Irregular triangle T(n, k), n >= 0, k = 1..A361757(n), read by rows; the n-th row lists the numbers k such that the Fibonacci numbers that appear in the dual Zeckendorf representation of k also appear in that of n.

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A361989 a(n) is the sum of the Fibonacci numbers missing from the dual Zeckendorf representation of n; a(0) = 0, and for n > 0, a(n) = A022290(A035327(A003754(n+1))).

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A361755 Irregular triangle T(n, k), n >= 0, k = 1..2^A007895(n), read by rows; the n-th row lists the numbers k such that the Fibonacci numbers that appear in the Zeckendorf representation of k also appear in that of n.

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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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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).

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A372659 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the Fibonacci numbers that appear in the Zeckendorf representation of n do not appear in the dual Zeckendorf representation of a(n).

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A356965 a(n) is the sum of the tribonacci numbers in common in the greedy and lazy tribonacci representations of n.

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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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