A343836 Array T(n, k), n, k > 0, read by antidiagonals; the balanced ternary representation of T(n, k) is obtained by adding componentwise (i.e., without carries) the digits in the balanced ternary representations of n and of k.

0, 1, 1, 2, -1, 2, 3, 3, 3, 3, 4, 4, -2, 4, 4, 5, 2, -4, -4, 2, 5, 6, 6, -3, -3, -3, 6, 6, 7, 7, 10, -2, -2, 10, 7, 7, 8, 5, 8, 8, -4, 8, 8, 5, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 13, 10, 10, -5, 10, 10, 13, 10, 10, 11, 8, 11, 11, 8, -7, -7, 8, 11, 11, 8, 11

Offset: 0

Author

Rémy Sigrist, May 01 2021

Keywords

• sign
• tabl
• base

Comments

This sequence is similar to A003987 and to A004489.

We use the following table to combine individual digits (this is the balanced ternary addition table read mod 3):

| T 0 1

---+-------

T | 1 T 0

0 | T 0 1

1 | 0 1 T

Examples

			Array T(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   4   2    6    7    5    9   10    8   12   13   11
    2|   2   3  -2  -4  -3   10    8    9   13   11   12    7    5    6
    3|   3   4  -4  -3  -2    8    9   10   11   12   13    5    6    7
    4|   4   2  -3  -2  -4    9   10    8   12   13   11    6    7    5
    5|   5   6  10   8   9   -5   -7   -6  -11  -13  -12   -8  -10   -9
    6|   6   7   8   9  10   -7   -6   -5  -13  -12  -11  -10   -9   -8
    7|   7   5   9  10   8   -6   -5   -7  -12  -11  -13   -9   -8  -10
    8|   8   9  13  11  12  -11  -13  -12   -8  -10   -9   -5   -7   -6
    9|   9  10  11  12  13  -13  -12  -11  -10   -9   -8   -7   -6   -5
   10|  10   8  12  13  11  -12  -11  -13   -9   -8  -10   -6   -5   -7
   11|  11  12   7   5   6   -8  -10   -9   -5   -7   -6  -11  -13  -12
   12|  12  13   5   6   7  -10   -9   -8   -7   -6   -5  -13  -12  -11
   13|  13  11   6   7   5   -9   -8  -10   -6   -5   -7  -12  -11  -13
Array T(n, k) begins in balanced ternary:
  n\k|    0    1   1T   10   11  1TT  1T0  1T1  10T  100  101  11T  110  111
  ---+----------------------------------------------------------------------
    0|    0    1   1T   10   11  1TT  1T0  1T1  10T  100  101  11T  110  111
    1|    1    T   10   11   1T  1T0  1T1  1TT  100  101  10T  110  111  11T
   1T|   1T   10   T1   TT   T0  101  10T  100  111  11T  110  1T1  1TT  1T0
   10|   10   11   TT   T0   T1  10T  100  101  11T  110  111  1TT  1T0  1T1
   11|   11   1T   T0   T1   TT  100  101  10T  110  111  11T  1T0  1T1  1TT
  1TT|  1TT  1T0  101  10T  100  T11  T1T  T10  TT1  TTT  TT0  T01  T0T  T00
  1T0|  1T0  1T1  10T  100  101  T1T  T10  T11  TTT  TT0  TT1  T0T  T00  T01
  1T1|  1T1  1TT  100  101  10T  T10  T11  T1T  TT0  TT1  TTT  T00  T01  T0T
  10T|  10T  100  111  11T  110  TT1  TTT  TT0  T01  T0T  T00  T11  T1T  T10
  100|  100  101  11T  110  111  TTT  TT0  TT1  T0T  T00  T01  T1T  T10  T11
  101|  101  10T  110  111  11T  TT0  TT1  TTT  T00  T01  T0T  T10  T11  T1T
  11T|  11T  110  1T1  1TT  1T0  T01  T0T  T00  T11  T1T  T10  TT1  TTT  TT0
  110|  110  111  1TT  1T0  1T1  T0T  T00  T01  T1T  T10  T11  TTT  TT0  TT1
  111|  111  11T  1T0  1T1  1TT  T00  T01  T0T  T10  T11  T1T  TT0  TT1  TTT
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10010
• Rémy Sigrist, Colored representation of the table for n, k < 1094 (blue denotes negative values, red denotes positive values, dark colors correspond to small values in absolute value)
• Wikipedia, Balanced ternary: Addition, subtraction and multiplication and division

Crossrefs

Cf. A003987, A004489, A343316.

Programs

PARI

T(n,k,c=v->centerlift(Mod(v,3))) = { if (n==0 && k==0, return (0), my (d=c(n), t=c(k)); c(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) = n.

T(n, n) = -n.