A333652 - OEIS

%I #30 Apr 02 2020 04:30:19

%S 1,1,3,1,6,17,17,6,1,10,45,167,404,570,460,186,1,15,100,506,2164,7726,

%T 20483,39401,56015,57632,37450,10340,1072,1,21,196,1316,7066,33983,

%U 147377,546400,1656592,4099732,8394433,14227675,19443270,20239262,14767415,7007270,1926990,230440

%N 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.

%H Seiichi Manyama, <a href="/A333652/b333652.txt">Rows n = 2..9, flattened</a>

%F T(n,0) = 1.

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

%e T(3,0) = 1;

%e    +--*

%e    |  |

%e    *  *

%e    |  |

%e    +--*

%e T(3,1) = 3;

%e    +--*--*   +--*--*   +--*

%e    |     |   |     |   |  |

%e    *     *   *  *--*   *  *--*

%e    |     |   |  |      |     |

%e    +--*--*   +--*      +--*--*

%e Triangle starts:

%e ====================================================================

%e n\k| 0   1    2     3      4 ...      7 ...  12 ...    17 ...    24

%e ---|----------------------------------------------------------------

%e 2  | 1;

%e 3  | 1,  3;

%e 4  | 1,  6,  17,   17,     6;

%e 5  | 1, 10,  45,  167,   404, ... , 186;

%e 6  | 1, 15, 100,  506,  2164, .......... , 1072;

%e 7  | 1, 21, 196, 1316,  7066, .................. , 230440;

%e 8  | 1, 28, 350, 3038, 20317, ............................ , 4638576;

%o (Python)

%o # Using graphillion

%o from graphillion import GraphSet

%o import graphillion.tutorial as tl

%o def A333652(n):

%o     universe = tl.grid(n - 1, n - 1)

%o     GraphSet.set_universe(universe)

%o     cycles = GraphSet.cycles().including(1).including(n)

%o     return [cycles.len(2 * k).len() for k in range(n, n * n // 2 + 1)]

%o print([i for n in range(2, 8) for i in A333652(n)])

%Y Row sums give A333247.

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

%K nonn,tabf

%O 2,3

%A _Seiichi Manyama_, Apr 01 2020