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