OFFSET
0,13
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).
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
FORMULA
EXAMPLE
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
PROG
(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)) }
CROSSREFS
KEYWORD
AUTHOR
Rémy Sigrist, Apr 11 2021
STATUS
approved