A339751 Number of (undirected) paths in the 3 X n king graph.

3, 235, 5148, 96956, 1622015, 25281625, 375341540, 5384233910, 75321922433, 1034169469257, 13999362291892, 187462552894846, 2489361245031701, 32843155609675341, 431132757745615932, 5637280548371484492, 73484574453020315121, 955615821857238062353, 12403944194214668554202

Offset: 1

Author

Seiichi Manyama, Dec 15 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..40
• Eric Weisstein's World of Mathematics, Graph Path
• Eric Weisstein's World of Mathematics, King Graph

Crossrefs

Row 3 of A307026.

Cf. A288527, A339761.

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 A339751(n):
    return A307026(n, 3)
print([A339751(n) for n in range(1, 21)])

Formula

Empirical g.f.: x*(3 + 142*x - 1234*x^2 + 6033*x^3 - 4437*x^4 + 1913*x^5 - 647*x^6 + 24874*x^7 + 25724*x^8 + 1737*x^9 + 10969*x^10 + 22767*x^11 + 24670*x^12 + 12330*x^13 + 1616*x^14 + 240*x^15 + 1008*x^16) / ((1 - x)^2 * (-1 + 8*x + 14*x^2 + 5*x^3 + 6*x^4)^2*(1 - 13*x - 2*x^2 + 45*x^3 - 24*x^4 - 22*x^5 + 9*x^6 + 8*x^7 - 6*x^8)). - Vaclav Kotesovec, Dec 16 2020