A339762 Number of (undirected) Hamiltonian paths in the 4 X n king graph.

1, 208, 4678, 171592, 4743130, 132202038, 3461461060, 88405359072, 2197293738684, 53565801482634, 1284136961473864, 30365618160010650, 709700882866473654, 16422374051280905778, 376744989106882359402, 8578133199326578887346, 194030408441913214687458

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

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 A339762(n):
    return B(n, 4)
print([A339762(n) for n in range(1, 11)])

Crossrefs

Row 4 of A350729.

Cf. A003695, A308129, A339760, A339761, A339763.

Sequence in context: A172967 A343507 A231111 * A223252 A223610 A184276

Adjacent sequences: A339759 A339760 A339761 * A339763 A339764 A339765

Keyword

nonn

Author

Seiichi Manyama, Dec 16 2020

Status

approved