A339198 Number of (undirected) cycles on the n X 4 king graph.

85, 3459, 136597, 4847163, 171903334, 6109759868, 217211571195, 7721452793328, 274480808918598, 9757216290644264, 346848710800215246, 12329747938579785459, 438296805656767232863, 15580536695961884270466, 553855562644922140772689, 19688409342958501534182423

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

Crossrefs

Column 4 of A339098.

Cf. A339201.

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 A339198(n):
    return A339098(n, 4)
print([A339198(n) for n in range(2, 20)])

Formula

Empirical g.f.: x^2 * (-336*x^16 - 360*x^15 + 187*x^14 - 4505*x^13 + 12123*x^12 + 14959*x^11 - 65728*x^10 + 50979*x^9 - 52680*x^8 + 26849*x^7 + 179877*x^6 + 22927*x^5 - 222548*x^4 + 1318*x^3 + 14878*x^2 + 399*x + 85) / ((x-1)^2 * (112*x^16 + 8*x^15 - 217*x^14 + 904*x^13 - 2866*x^12 + 1756*x^11 + 7818*x^10 - 22167*x^9 + 45698*x^8 - 61238*x^7 + 8041*x^6 + 31909*x^5 - 5819*x^4 - 538*x^3 - 36*x^2 - 34*x + 1)). - Vaclav Kotesovec, Dec 09 2020