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.

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

Author

Rémy Sigrist, Apr 24 2020

Keywords

• nonn
• tabl
• look
• base

Comments

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

Examples

			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
		

Links

• 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

Crossrefs

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

Programs

PARI

See Links section.

Formula

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).