A343316 Array T(n, k), n, k > 0, read by antidiagonals; the balanced ternary representation of T(n, k) is obtained by multiplying componentwise the digits in the balanced ternary representations of n and of k.

0, 0, 0, 0, 1, 0, 0, -1, -1, 0, 0, 0, 4, 0, 0, 0, 1, 3, 3, 1, 0, 0, -1, 2, 3, 2, -1, 0, 0, 0, -2, 3, 3, -2, 0, 0, 0, 1, -3, -3, 4, -3, -3, 1, 0, 0, -1, -4, -3, -4, -4, -3, -4, -1, 0, 0, 0, 1, -3, -3, 13, -3, -3, 1, 0, 0, 0, 1, 0, 0, -2, 12, 12, -2, 0, 0, 1, 0

Offset: 0

Author

Rémy Sigrist, Apr 11 2021

Keywords

• sign
• tabl
• base

Comments

For any k >= 0, n -> T(n, k) is 3^A134021(k)-periodic.

The zeros of the table form a Vicsek fractal (see illustration in Links section).

Examples

			Array T(n, k) begins:
  n\k|  0   1   2   3   4   5   6   7   8  9  10  11  12
  ---+--------------------------------------------------
    0|  0   0   0   0   0   0   0   0   0  0   0   0   0
    1|  0   1  -1   0   1  -1   0   1  -1  0   1  -1   0
    2|  0  -1   4   3   2  -2  -3  -4   1  0  -1   4   3
    3|  0   0   3   3   3  -3  -3  -3   0  0   0   3   3
    4|  0   1   2   3   4  -4  -3  -2  -1  0   1   2   3
    5|  0  -1  -2  -3  -4  13  12  11  10  9   8   7   6
    6|  0   0  -3  -3  -3  12  12  12   9  9   9   6   6
    7|  0   1  -4  -3  -2  11  12  13   8  9  10   5   6
    8|  0  -1   1   0  -1  10   9   8  10  9   8  10   9
    9|  0   0   0   0   0   9   9   9   9  9   9   9   9
   10|  0   1  -1   0   1   8   9  10   8  9  10   8   9
   11|  0  -1   4   3   2   7   6   5  10  9   8  13  12
   12|  0   0   3   3   3   6   6   6   9  9   9  12  12
Array T(n, k) begins in balanced ternary notation (with "T" instead of digits "-1"):
  n\k|  0  1  1T  10  11  1TT  1T0  1T1  10T  100  101  11T  110
  ---+----------------------------------------------------------
    0|  0  0   0   0   0    0    0    0    0    0    0    0    0
    1|  0  1   T   0   1    T    0    1    T    0    1    T    0
   1T|  0  T  11  10  1T   T1   T0   TT    1    0    T   11   10
   10|  0  0  10  10  10   T0   T0   T0    0    0    0   10   10
   11|  0  1  1T  10  11   TT   T0   T1    T    0    1   1T   10
  1TT|  0  T  T1  T0  TT  111  110  11T  101  100  10T  1T1  1T0
  1T0|  0  0  T0  T0  T0  110  110  110  100  100  100  1T0  1T0
  1T1|  0  1  TT  T0  T1  11T  110  111  10T  100  101  1TT  1T0
  10T|  0  T   1   0   T  101  100  10T  101  100  10T  101  100
  100|  0  0   0   0   0  100  100  100  100  100  100  100  100
  101|  0  1   T   0   1  10T  100  101  10T  100  101  10T  100
  11T|  0  T  11  10  1T  1T1  1T0  1TT  101  100  10T  111  110
  110|  0  0  10  10  10  1T0  1T0  1T0  100  100  100  110  110
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10010
• Rémy Sigrist, Colored representation of the table for n, k < 3^6 (where the color denotes the sign of T(n, k): red for positive values, blue for negative values, white for zeros)
• Wikipedia, Viczek fractal

Crossrefs

Cf. A059095, A060374, A102283, A134021, A343317.

Programs

PARI

T(n,k) = { if (n==0 || k==0, return (0), my (d=centerlift(Mod(n,3)), t=centerlift(Mod(k,3))); d*t + 3*T((n-d)\3, (k-t)\3)) }

Formula

T(n, k) = T(k, n).

T(m, T(n, k)) = T(T(m, n), k).

T(n, 0) = 0.

T(n, 1) = A102283(n).

T(n, n) = A060374(n).