A334348 -id:A334348 - cp's OEIS frontend

A356325 Array A(n, k), n, k >= 0, read by antidiagonals; the terms in the negaFibonacci representation of A(n, k) are the terms in common in the negaFibonacci representations of n and k.

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

Offset: 0

Rémy Sigrist, Aug 03 2022

This sequence has similarities with A334348.

			Array A(n, k) 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  1   0  0  1  0  1   0   0   1   0   0
    2|  0  0  2  2   0  0  0  2  2   0   0   0   0   0
    3|  0  1  2  3   0  0  1  2  3   0   0   1   0   0
    4|  0  0  0  0   4  5  5  5  5  -1   0   0  -1   0
    5|  0  0  0  0   5  5  5  5  5   0   0   0   0   0
    6|  0  1  0  1   5  5  6  5  6   0   0   1   0   0
    7|  0  0  2  2   5  5  5  7  7   0   0   0   0   0
    8|  0  1  2  3   5  5  6  7  8   0   0   1   0   0
    9|  0  0  0  0  -1  0  0  0  0   9  10  10  12  13
   10|  0  0  0  0   0  0  0  0  0  10  10  10  13  13
   11|  0  1  0  1   0  0  1  0  1  10  10  11  13  13
   12|  0  0  0  0  -1  0  0  0  0  12  13  13  12  13
   13|  0  0  0  0   0  0  0  0  0  13  13  13  13  13
.
For n = 14 and k = 43:
- using F(-k) = A039834(k):
- 14 = F(-1) + F(-7),
- 43 = F(-2) + F(-4) + F(-7) + F(-9),
- so A(14, 43) = F(-7) = 13.

Rémy Sigrist, Colored representation of the array for n, k <= 1000 (white for 0's, shades of blue for negative values, shades of red for positive values)
Rémy Sigrist, PARI program
Wikipedia, NegaFibonacci coding
Index entries for sequences related to Zeckendorf expansion of n

Cf. A004198, A039834, A215024, A309076, A334348, A356326, A356327.

PARI
```
See Links section.
```

A(n, k) = A(k, n).
A(n, n) = n.
A(n, 0) = 0.
A(n, k) = A356327(A215024(n) AND A215024(k)) (where AND denotes the bitwise AND operator).

A356969 A(n, k) is the sum of the terms in common in the dual Zeckendorf representations of n and of k; square array A(n, k) read by antidiagonals, n, k >= 0.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Sep 06 2022

The dual Zeckendorf representation corresponds to the lazy Fibonacci representation.

See A334348 for the sequence dealing with Zeckendorf (or greedy Fibonacci) representations. Unlike A334348, the present sequence is not associative.

			Square array A(n, k) 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  1  1  0  1  0  1  1   0   1   1   0
    2|  0  0  2  2  0  2  2  2  2  0   2   2   0   2
    3|  0  1  2  3  1  2  3  2  3  1   2   3   1   2
    4|  0  1  0  1  4  3  4  0  1  4   3   4   4   3
    5|  0  0  2  2  3  5  5  2  2  3   5   5   3   5
    6|  0  1  2  3  4  5  6  2  3  4   5   6   4   5
    7|  0  0  2  2  0  2  2  7  7  5   7   7   0   2
    8|  0  1  2  3  1  2  3  7  8  6   7   8   1   2
    9|  0  1  0  1  4  3  4  5  6  9   8   9   4   3
   10|  0  0  2  2  3  5  5  7  7  8  10  10   3   5
   11|  0  1  2  3  4  5  6  7  8  9  10  11   4   5
   12|  0  1  0  1  4  3  4  0  1  4   3   4  12  11
   13|  0  0  2  2  3  5  5  2  2  3   5   5  11  13

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

Cf. A003754, A022290, A334348.

PARI
```
See Links section.
```

A(n, k) = A022290(A003754(n+1) AND A003754(k+1)) (where AND denotes the bitwise AND operator, A004198).
A(n, k) = A(k, n).
A(n, 0) = 0.
A(n, n) = n.

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
```
See Links section.
```

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

A356325 Array A(n, k), n, k >= 0, read by antidiagonals; the terms in the negaFibonacci representation of A(n, k) are the terms in common in the negaFibonacci representations of n and k.

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A356969 A(n, k) is the sum of the terms in common in the dual Zeckendorf representations of n and of k; square array A(n, k) read by antidiagonals, n, k >= 0.

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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.

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Keywords

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Examples

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Crossrefs

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