A333668 Triangle T(n,k), n >= 2, 0 <= k <= floor(n^2/2)-2*n+2, read by rows, where T(n,k) is the number of 2*(k+2*n-2)-cycles in the n X n grid graph which pass through four corners ((0,0), (0,n-1), (n-1,n-1), (n-1,0)).

1, 1, 1, 4, 6, 1, 12, 58, 156, 146, 1, 24, 244, 1416, 5435, 12976, 16654, 7108, 1072, 1, 40, 696, 7076, 47965, 236628, 873610, 2348664, 4335724, 4958224, 3407276, 1298704, 205792

Offset: 2

Author

Seiichi Manyama, Apr 01 2020

Keywords

• nonn
• tabf

Examples

			T(4,1) = 4;
   +--*--*--+   +--*--*--+   +--*--*--+   +--*  *--+
   |        |   |        |   |        |   |  |  |  |
   *--*     *   *     *--*   *        *   *  *--*  *
      |     |   |     |      |        |   |        |
   *--*     *   *     *--*   *  *--*  *   *        *
   |        |   |        |   |  |  |  |   |        |
   +--*--*--+   +--*--*--+   +--*  *--+   +--*--*--+
Triangle starts:
=================================================================
n\k| 0   1     2      3       4 ...       8 ...    12 ...     18
---|-------------------------------------------------------------
2  | 1;
3  | 1;
4  | 1,  4,    6;
5  | 1, 12,   58,   156,    146;
6  | 1, 24,  244,  1416,   5435, ... , 1072;
7  | 1, 40,  696,  7076,  47965, ........... , 205792;
8  | 1, 60, 1590, 24960, 263770, ..................... , 4638576;
		

Links

• Seiichi Manyama, Rows n = 2..9, flattened

Crossrefs

Row sums give A333466.

Cf. A003763, A046092, A302337, A333651, A333652, A333667.

Programs

Python

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333668(n):
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles()
    for i in [1, n, n * (n - 1) + 1, n * n]:
        cycles = cycles.including(i)
    return [cycles.len(2 * k).len() for k in range(2 * n - 2, n * n // 2 + 1)]
print([i for n in range(2, 8) for i in A333668(n)])

Formula

T(n,0) = 1.

T(n,1) = A046092(n-3).