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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).

%I A333651 #39 Apr 01 2020 09:56:32
%S A333651 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,
%T A333651 20,74,298,1228,4762,15984,40026,75524,109150,121130,99032,51964,
%U A333651 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
%N 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).
%H A333651 Seiichi Manyama, <a href="/A333651/b333651.txt">Rows n = 2..9, flattened</a>
%F A333651 T(n,k) = A034010(k+2) for k <= n-2.
%e A333651 T(3,0) = 1;
%e A333651    +--*
%e A333651    |  |
%e A333651    *--*
%e A333651 T(3,1) = 2;
%e A333651    +--*--*   +--*
%e A333651    |     |   |  |
%e A333651    *--*--*   *  *
%e A333651              |  |
%e A333651              *--*
%e A333651 T(3,2) = 4;
%e A333651    +--*--*   +--*--*   +--*--*   +--*
%e A333651    |     |   |     |   |     |   |  |
%e A333651    *     *   *  *--*   *--*  *   *  *--*
%e A333651    |     |   |  |         |  |   |     |
%e A333651    *--*--*   *--*         *--*   *--*--*
%e A333651 Triangle starts:
%e A333651 ===================================================
%e A333651 n\k| 0  1  2   3   4    5     6 ...     10 ...  16
%e A333651 ---|-----------------------------------------------
%e A333651 2  | 1;
%e A333651 3  | 1, 2, 4;
%e A333651 4  | 1, 2, 6, 18, 40,  24,    6;
%e A333651 5  | 1, 2, 6, 20, 72, 248,  698, ... , 206;
%e A333651 6  | 1, 2, 6, 20, 74, 298, 1228, .......... , 1072;
%e A333651 7  | 1, 2, 6, 20, 74, 300, 1300, ...
%e A333651 8  | 1, 2, 6, 20, 74, 300, 1302, ...
%e A333651 9  | 1, 2, 6, 20, 74, 300, 1302, ...
%o A333651 (Python)
%o A333651 # Using graphillion
%o A333651 from graphillion import GraphSet
%o A333651 import graphillion.tutorial as tl
%o A333651 def A333651(n):
%o A333651     universe = tl.grid(n - 1, n - 1)
%o A333651     GraphSet.set_universe(universe)
%o A333651     cycles = GraphSet.cycles().including(1)
%o A333651     return [cycles.len(2 * k).len() for k in range(2, n * n // 2 + 1)]
%o A333651 print([i for n in range(2, 8) for i in A333651(n)])
%Y A333651 Row sums give A333246.
%Y A333651 Cf. A003763, A034010, A302337, A333652, A333667, A333668.
%K A333651 nonn,tabf
%O A333651 2,3
%A A333651 _Seiichi Manyama_, Apr 01 2020