A339098 Square array T(n,k), n >= 2, k >= 2, read by antidiagonals, where T(n,k) is the number of (undirected) cycles on the n X k king graph.

7, 30, 30, 85, 348, 85, 204, 3459, 3459, 204, 451, 33145, 136597, 33145, 451, 954, 316164, 4847163, 4847163, 316164, 954, 1969, 3013590, 171903334, 545217435, 171903334, 3013590, 1969, 4008, 28722567, 6109759868, 61575093671, 61575093671, 6109759868, 28722567, 4008

Offset: 2

Author

Seiichi Manyama, Nov 27 2020

Keywords

• nonn
• tabl

Examples

			Square array T(n,k) begins:
    7,     30,        85,         204,            451, ...
   30,    348,      3459,       33145,         316164, ...
   85,   3459,    136597,     4847163,      171903334, ...
  204,  33145,   4847163,   545217435,    61575093671, ...
  451, 316164, 171903334, 61575093671, 21964731190911, ...
		

Links

• Eric Weisstein's World of Mathematics, Graph Cycle
• Eric Weisstein's World of Mathematics, King Graph

Crossrefs

Rows and columns 2..5 give A339196, A339197, A339198, A339199.

Main diagonal gives A234622.

Cf. A231829, A339190.

Programs

Python

# Using graphillion
from graphillion import GraphSet
def make_nXk_king_graph(n, k):
    grids = []
    for i in range(1, k + 1):
        for j in range(1, n):
            grids.append((i + (j - 1) * k, i + j * k))
            if i < k:
                grids.append((i + (j - 1) * k, i + j * k + 1))
            if i > 1:
                grids.append((i + (j - 1) * k, i + j * k - 1))
    for i in range(1, k * n, k):
        for j in range(1, k):
            grids.append((i + j - 1, i + j))
    return grids
def A339098(n, k):
    universe = make_nXk_king_graph(n, k)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles()
    return cycles.len()
print([A339098(j + 2, i - j + 2) for i in range(9 - 1) for j in range(i + 1)])

Formula

T(n,k) = T(k,n).