OFFSET
1,1
COMMENTS
Row sums are equal to n for even n and (n-1)/2 for odd n; or A065423(n+1).
FORMULA
T(n, k) = 1/2 + (1/2)*(-1)^(n*(k+1)), for n >= 1 and 0 <= k <= n-1.
T(n, k) = (2^n - 2^(n-k-1) - 2^k) mod 3, for n >= 1 and 0 <= k <= n-1.
T(n, k) = A358125(n, k) mod 3, for n >= 1 and 0 <= k <= n-1.
EXAMPLE
Triangle begins:
n\k 0 1 2 3 4 5 6 7 8 9 ...
1 0;
2 1, 1;
3 0, 1, 0;
4 1, 1, 1, 1;
5 0, 1, 0, 1, 0;
6 1, 1, 1, 1, 1, 1;
7 0, 1, 0, 1, 0, 1, 0;
8 1, 1, 1, 1, 1, 1, 1, 1;
9 0, 1, 0, 1, 0, 1, 0, 1, 0;
10 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
...
Formatted as a symmetric triangle -- regular hexagram pattern with 0's at the centers formed by connecting all 1's:
.----------------------------------------------.
| k=0 1 2 3 4 5 |
|-----------------------/---/---/---/---/--./ |
------- / / / / / |
| n=1 | 0 / / / / /| |
------- / / / / | 6 |
| 2 | 1---1 / / / / |/ |
------- \ / / / / / |
| 3 | 0 1 0 / / / /| |
------- / \ / / / | 7 |
| 4 | 1---1---1---1 / / / |/ |
------- \ / \ / / / / |
| 5 | 0 1 0 1 0 / / /| |
------- / \ / \ / / | 8 |
| 6 | 1---1---1---1---1---1 / / |/ |
------- \ / \ / \ / / / |
| 7 | 0 1 0 1 0 1 0 / /| |
------- / \ / \ / \ / | 9 |
| 8 | 1---1---1---1---1---1---1---1 / / |
------- \ / \ / \ / \ / / |
| 9 | 0 1 0 1 0 1 0 1 0 /| |
------- / \ / \ / \ / \ | . |
| 10 | 1---1---1---1---1---1---1---1---1---1 | . |
------- | . |
MAPLE
T := n -> local k; seq(1/2 + 1/2*(-1)^(n*(k + 1)), k = 0 .. n - 1); # formula 1
seq(T(n), n=1..16); # print first 16 rows of formula 1.
PROG
(PARI) T(n, k) = bitnegimply(1, n) || bitand(1, k); \\ Kevin Ryde, Dec 21 2022
CROSSREFS
KEYWORD
AUTHOR
Ambrosio Valencia-Romero, Dec 20 2022
STATUS
approved