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

0,4

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

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

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.

Example

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

Prog

(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)) }

Crossrefs

Cf. A003987, A004489, A343316.

Sequence in context: A343040 A343044 A003986 * A123603 A228506 A228285

Adjacent sequences: A343833 A343834 A343835 * A343837 A343838 A343839

Keyword

sign,tabl,base

Author

Rémy Sigrist, May 01 2021

Status

approved