A339197 Number of (undirected) cycles on the n X 3 king graph.

30, 348, 3459, 33145, 316164, 3013590, 28722567, 273751765, 2609096478, 24866992602, 237004387635, 2258860992595, 21528938911842, 205189789087374, 1955639788756293, 18638973217791295, 177645865363829526, 1693121885638023396, 16136945905019298321, 153799336805212613275

Offset: 2

Author

Seiichi Manyama, Nov 27 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 2..1000
• Eric Weisstein's World of Mathematics, Graph Cycle
• Eric Weisstein's World of Mathematics, King Graph

Crossrefs

Column 3 of A339098.

Cf. A339200.

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()
def A339197(n):
    return A339098(n, 3)
print([A339197(n) for n in range(2, 30)])

Formula

Empirical g.f.: -x^2 * (11*x^4 + 49*x^3 + 69*x^2 + 48*x + 30) / ((x-1)^2 * (6*x^4 + 5*x^3 + 14*x^2 + 8*x - 1)). - Vaclav Kotesovec, Dec 09 2020