A339201 Number of (undirected) Hamiltonian cycles on the n X 4 king graph.

8, 120, 2830, 50354, 1003218, 19380610, 378005474, 7348400816, 143013145124, 2782280184314, 54134923232608, 1053263634537410, 20492847566047336, 398717839924458408, 7757640305938339162, 150936198726479633524, 2936684182444832427774, 57137476790772843457886

Offset: 2

Author

Seiichi Manyama, Nov 27 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 2..500
• 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 4 of A339190.

Cf. A339198.

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 A339201(n):
    return A339190(n, 4)
print([A339201(n) for n in range(2, 20)])

Formula

Empirical g.f.: 2*x^2 * (56*x^16 + 53*x^15 + 413*x^14 - 943*x^13 - 635*x^12 - 700*x^11 + 2283*x^10 + 455*x^9 + 3044*x^8 - 4856*x^7 - 4293*x^6 + 6475*x^5 + 719*x^4 - 1386*x^3 + 143*x^2 - 8*x + 4) / (112*x^16 + 106*x^15 + 964*x^14 - 1933*x^13 + 357*x^12 - 3503*x^11 + 3756*x^10 - 828*x^9 + 12662*x^8 - 18201*x^7 - 2441*x^6 + 5486*x^5 - 704*x^4 + 318*x^3 - 63*x^2 - 17*x + 1). - Vaclav Kotesovec, Dec 09 2020