A333571 - OEIS

%I #14 Mar 27 2020 11:57:38

%S 1,1,2,1,2,4,1,2,8,6,1,2,16,14,10,1,2,32,34,38,14,1,2,64,80,162,74,20,

%T 1,2,128,190,650,426,170,26,1,2,256,450,2728,2166,1594,338,34,1,2,512,

%U 1066,11250,12014,12908,4374,724,42,1,2,1024,2526,46984,62714,119364,47738,14640,1448,52

%N Square array T(n,k), n >= 1, k >= 2, read by antidiagonals, where T(n,k) is the number of Hamiltonian paths in the n X k grid graph which start at any of the n vertices on left side of the graph and terminate at any of the n vertices on the right side.

%H Seiichi Manyama, <a href="/A333571/b333571.txt">Antidiagonals n = 1..14, flattened</a>

%e Square array T(n,k) begins:

%e    1,  1,   1,    1,     1,     1,      1, ...

%e    2,  2,   2,    2,     2,     2,      2, ...

%e    4,  8,  16,   32,    64,   128,    256, ...

%e    6, 14,  34,   80,   190,   450,   1066, ...

%e   10, 38, 162,  650,  2728, 11250,  46984, ...

%e   14, 74, 426, 2166, 12014, 62714, 340510, ...

%o (Python)

%o # Using graphillion

%o from graphillion import GraphSet

%o import graphillion.tutorial as tl

%o def A(start, goal, n, k):

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

%o     GraphSet.set_universe(universe)

%o     paths = GraphSet.paths(start, goal, is_hamilton=True)

%o     return paths.len()

%o def A333571(n, k):

%o     if n == 1: return 1

%o     s = 0

%o     for i in range(1, n + 1):

%o         for j in range(k * n - n + 1, k * n + 1):

%o             s += A(i, j, k, n)

%o     return s

%o print([A333571(j + 1, i - j + 2) for i in range(11) for j in range(i + 1)])

%Y Columns k=2-3 give: A333574, A333575.

%Y Rows n=1-3 give: A000012, 2*A000012, A000079.

%Y T(n,n) gives A121789(n-1).

%Y Cf. A333509.

%K nonn,tabl

%O 1,3

%A _Seiichi Manyama_, Mar 27 2020