A339761 Number of (undirected) Hamiltonian paths in the 3 X n king graph.

1, 48, 392, 4678, 43676, 406396, 3568906, 30554390, 254834078, 2085479610, 16791859330, 133416458104, 1048095087616, 8154539310958, 62918331433308, 481954854686434, 3668399080453520, 27766093432542984, 209120844634276158, 1568050593805721822

Offset: 1

Author

Seiichi Manyama, Dec 16 2020

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• Eric Weisstein's World of Mathematics, Graph Path
• Eric Weisstein's World of Mathematics, King Graph
• Index entries for linear recurrences with constant coefficients, signature (15,-36,-289,708,2617,-1278,-4641,2263,4808,3286,-1422,-3830,-2200, -432,216,216).

Crossrefs

Row 3 of A350729.

Cf. A003685, A308129, A339751, A339760, A339762, A339763.

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 A(start, goal, n, k):
    universe = make_nXk_king_graph(n, k)
    GraphSet.set_universe(universe)
    paths = GraphSet.paths(start, goal, is_hamilton=True)
    return paths.len()
def B(n, k):
    m = k * n
    s = 0
    for i in range(1, m):
        for j in range(i + 1, m + 1):
            s += A(i, j, n, k)
    return s
def A339761(n):
    return B(n, 3)
print([A339761(n) for n in range(1, 11)])

Formula

G.f.: x*(1 + 33*x - 292*x^2 + 815*x^3 + 782*x^4 - 3649*x^5 - 4630*x^6 + 1517*x^7 + 3835*x^8 - 3822*x^9 - 5722*x^10 - 5418*x^11 - 7562*x^12 - 4808*x^13 - 240*x^14 + 720*x^15 + 216*x^16)/((1 - x)*(1 - 4*x - 15*x^2 - 8*x^3 - 6*x^4)^2*(1 - 6*x - 12*x^2 + 27*x^3 - 2*x^4 - 30*x^5 - 4*x^6 + 6*x^7)). - Andrew Howroyd, Jan 17 2022