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

2,1

Links

Table of n, a(n) for n=2..37.

Eric Weisstein's World of Mathematics, Graph Cycle

Eric Weisstein's World of Mathematics, King Graph

Formula

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

Example

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, ...

Prog

(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)])

Crossrefs

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

Main diagonal gives A234622.

Cf. A231829, A339190.

Sequence in context: A157422 A061644 A053621 * A210107 A266604 A018831

Adjacent sequences: A339095 A339096 A339097 * A339099 A339100 A339101

Keyword

nonn,tabl

Author

Seiichi Manyama, Nov 27 2020

Status

approved