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.

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

Author

Rémy Sigrist, Sep 06 2022

Keywords

• nonn
• base
• tabl

Comments

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.

Examples

			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
		

Links

• 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

Crossrefs

Cf. A003754, A022290, A334348.

Programs

PARI

See Links section.

Formula

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.