A343602 For any positive number n, the balanced ternary representation of a(n) is obtained by left-rotating the balanced ternary representation of n until a nonzero digit appears again as the leftmost digit; a(0) = 0.

0, 1, -2, 3, 4, -11, -8, -5, -6, 9, 12, 7, 10, 13, -38, -35, -32, -29, -26, -23, -20, -17, -14, -33, -24, -15, -18, 27, 36, 21, 30, 39, 16, 19, 22, 25, 28, 31, 34, 37, 40, -119, -116, -113, -110, -107, -104, -101, -98, -95, -92, -89, -86, -83, -80, -77, -74

Offset: 0

Author

Rémy Sigrist, Apr 21 2021

Keywords

• sign
• base

Comments

This sequence can be extended to negative indexes by setting a(-n) = -a(n) for any n > 0. We then obtain a permutation of the integers (Z) with inverse A343601 (after a similar extension to negative indexes).

Examples

			The first terms, in base 10 and in balanced ternary (where T denotes the digit -1), are:
  n   a(n)  bter(n)  bter(a(n))
  --  ----  -------  ----------
   0     0        0           0
   1     1        1           1
   2    -2       1T          T1
   3     3       10          10
   4     4       11          11
   5   -11      1TT         TT1
   6    -8      1T0         T01
   7    -5      1T1         T11
   8    -6      10T         T10
   9     9      100         100
  10    12      101         110
  11     7      11T         1T1
  12    10      110         101
  13    13      111         111
  14   -38     1TTT        TTT1
  15   -35     1TT0        TT01
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..9841

Crossrefs

Cf. A005812, A065363, A134021, A139708 (binary variant), A343600 (ternary variant), A343603 (inverse).

Programs

PARI

a(n) = { my (d = [], t); while (n, d = concat(t = centerlift(Mod(n,3)), d); n = (n-t)\3); for (k=2, #d, if (d[k], return (fromdigits(concat(d[k..#d], d[1..k-1]), 3)))); return (fromdigits(d, 3)) }

Formula

A065363(a(n)) = A065363(n).

A134021(a(n)) = A134021(n).

a^k(n) = n for k = A005812(n) (where a^k denotes the k-th iterate of a).