A333651 Triangle T(n,k), n >= 2, 0 <= k <= floor(n^2/2)-2, read by rows, where T(n,k) is the number of 2*(k+2)-cycles in the n X n grid graph which pass through NW corner (0,0).

1, 1, 2, 4, 1, 2, 6, 18, 40, 24, 6, 1, 2, 6, 20, 72, 248, 698, 1100, 1096, 662, 206, 1, 2, 6, 20, 74, 298, 1228, 4762, 15984, 40026, 75524, 109150, 121130, 99032, 51964, 11996, 1072, 1, 2, 6, 20, 74, 300, 1300, 5844, 26148, 110942, 427388, 1393796, 3790524, 8648638, 16727776, 27529284, 38120312, 43012614, 37385280, 23166526, 9496426, 2286972, 242764

Offset: 2

Author

Seiichi Manyama, Apr 01 2020

Keywords

• nonn
• tabf

Examples

			T(3,0) = 1;
   +--*
   |  |
   *--*
T(3,1) = 2;
   +--*--*   +--*
   |     |   |  |
   *--*--*   *  *
             |  |
             *--*
T(3,2) = 4;
   +--*--*   +--*--*   +--*--*   +--*
   |     |   |     |   |     |   |  |
   *     *   *  *--*   *--*  *   *  *--*
   |     |   |  |         |  |   |     |
   *--*--*   *--*         *--*   *--*--*
Triangle starts:
===================================================
n\k| 0  1  2   3   4    5     6 ...     10 ...  16
---|-----------------------------------------------
2  | 1;
3  | 1, 2, 4;
4  | 1, 2, 6, 18, 40,  24,    6;
5  | 1, 2, 6, 20, 72, 248,  698, ... , 206;
6  | 1, 2, 6, 20, 74, 298, 1228, .......... , 1072;
7  | 1, 2, 6, 20, 74, 300, 1300, ...
8  | 1, 2, 6, 20, 74, 300, 1302, ...
9  | 1, 2, 6, 20, 74, 300, 1302, ...
		

Links

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

Crossrefs

Row sums give A333246.

Cf. A003763, A034010, A302337, A333652, A333667, A333668.

Programs

Python

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333651(n):
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles().including(1)
    return [cycles.len(2 * k).len() for k in range(2, n * n // 2 + 1)]
print([i for n in range(2, 8) for i in A333651(n)])

Formula

T(n,k) = A034010(k+2) for k <= n-2.