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

2,3

Links

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

Formula

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

Example

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, ...

Prog

(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)])

Crossrefs

Row sums give A333246.

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

Sequence in context: A194671 A331696 A364837 * A131398 A249307 A209581

Adjacent sequences: A333648 A333649 A333650 * A333652 A333653 A333654

Keyword

nonn,tabf

Author

Seiichi Manyama, Apr 01 2020

Status

approved