A339750 Number of (undirected) paths in the 2 X n king graph.

1, 30, 235, 1448, 7909, 40674, 202719, 994268, 4837337, 23441366, 113377235, 547864528, 2646278093, 12779454410, 61709221831, 297968336836, 1438739595201, 6946894643134, 33542671171515, 161958548471736, 782005482553269, 3775857399168946, 18231454211243951, 88029252078796716

Offset: 1

Author

Seiichi Manyama, Dec 15 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..50
• Eric Weisstein's World of Mathematics, Graph Path
• Eric Weisstein's World of Mathematics, King Graph

Crossrefs

Row 2 of A307026.

Cf. A288516, A339760.

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)
    return paths.len()
def A307026(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 A339750(n):
    return A307026(n, 2)
print([A339750(n) for n in range(1, 21)])

Formula

Empirical g.f.: x*(16*x^4 - 48*x^3 + 32*x^2 - 20*x - 1) / ((x-1)^2 * (2*x - 1)^2 * (4*x^2 + 4*x - 1)). - Vaclav Kotesovec, Dec 16 2020