A339200 Number of (undirected) Hamiltonian cycles on the n X 3 king graph.

4, 16, 120, 744, 4922, 31904, 208118, 1354872, 8826022, 57483536, 374412158, 2438639080, 15883563110, 103454037120, 673825180718, 4388811619032, 28585557862518, 186185731404016, 1212679737590398, 7898522254036168, 51445284278407878, 335077523213321312, 2182453613487235150, 14214930709900240312

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, Hamiltonian Cycle
• Eric Weisstein's World of Mathematics, King Graph
• Index entries for sequences related to graphs, Hamiltonian

Crossrefs

Column 3 of A339190.

Cf. A339197.

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 A339190(n, k):
    universe = make_nXk_king_graph(n, k)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles(is_hamilton=True)
    return cycles.len()
def A339200(n):
    return A339190(n, 3)
print([A339200(n) for n in range(2, 20)])

Formula

Empirical g.f.: 2*x^2 * (3*x^4 + 4*x^3 + 2*x^2 - 2) / (6*x^4 + 8*x^3 + 15*x^2 + 4*x - 1). - Vaclav Kotesovec, Dec 09 2020