A357298 Triangle read by rows where all entries in every even row are 1's and the entries in every odd row alternate between 0 (start/end) and 1.

0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

1,1

Comments

Row sums are equal to n for even n and (n-1)/2 for odd n; or A065423(n+1).

Links

Table of n, a(n) for n=1..136.

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

Cf. A358125, A065423 (row sums).

Sequence in context: A144101 A127000 A103226 * A080886 A217207 A083924

Adjacent sequences: A357295 A357296 A357297 * A357299 A357300 A357301

Keyword

nonn,easy,tabl

Author

Ambrosio Valencia-Romero, Dec 20 2022

Status

approved