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

1,2

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).

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

Prog

(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)])

Crossrefs

Row 3 of A350729.

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

Sequence in context: A348120 A160335 A019558 * A232917 A172201 A249293

Adjacent sequences: A339758 A339759 A339760 * A339762 A339763 A339764

Keyword

nonn

Author

Seiichi Manyama, Dec 16 2020

Status

approved