A333652 Triangle T(n,k), n >= 2, 0 <= k <= floor(n^2/2)-n, read by rows, where T(n,k) is the number of 2*(k+n)-cycles in the n X n grid graph which pass through NW and SW corners.

1, 1, 3, 1, 6, 17, 17, 6, 1, 10, 45, 167, 404, 570, 460, 186, 1, 15, 100, 506, 2164, 7726, 20483, 39401, 56015, 57632, 37450, 10340, 1072, 1, 21, 196, 1316, 7066, 33983, 147377, 546400, 1656592, 4099732, 8394433, 14227675, 19443270, 20239262, 14767415, 7007270, 1926990, 230440

Offset: 2

Author

Seiichi Manyama, Apr 01 2020

Keywords

• nonn
• tabf

Examples

			T(3,0) = 1;
   +--*
   |  |
   *  *
   |  |
   +--*
T(3,1) = 3;
   +--*--*   +--*--*   +--*
   |     |   |     |   |  |
   *     *   *  *--*   *  *--*
   |     |   |  |      |     |
   +--*--*   +--*      +--*--*
Triangle starts:
====================================================================
n\k| 0   1    2     3      4 ...      7 ...  12 ...    17 ...    24
---|----------------------------------------------------------------
2  | 1;
3  | 1,  3;
4  | 1,  6,  17,   17,     6;
5  | 1, 10,  45,  167,   404, ... , 186;
6  | 1, 15, 100,  506,  2164, .......... , 1072;
7  | 1, 21, 196, 1316,  7066, .................. , 230440;
8  | 1, 28, 350, 3038, 20317, ............................ , 4638576;
		

Links

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

Crossrefs

Row sums give A333247.

Cf. A000217, A003763, A302337, A333651, A333667, A333668.

Programs

Python

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333652(n):
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles().including(1).including(n)
    return [cycles.len(2 * k).len() for k in range(n, n * n // 2 + 1)]
print([i for n in range(2, 8) for i in A333652(n)])

Formula

T(n,0) = 1.

T(n,1) = A000217(n-1) for n > 2.