A338709 Number of (undirected) paths in C_3 X P_n.

6, 129, 1209, 8856, 57522, 348945, 2031525, 11531712, 64438638, 356590161, 1961459841, 10749416568, 58777575354, 320956083777, 1751147966157, 9549634751424, 52062358139670, 283782668909793, 1546691543230473, 8429380058864280, 45938035123043586, 250345837703068209

Offset: 1

Author

Seiichi Manyama, Dec 18 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..50

Crossrefs

Cf. A003689, A338960, A338961, A338962, A338963.

Programs

Python

# Using graphillion
from graphillion import GraphSet
def make_CnXPk(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))
        grids.append((i + (n - 1) * k, i))
    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_CnXPk(n, k)
    GraphSet.set_universe(universe)
    paths = GraphSet.paths(start, goal)
    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 A338709(n):
    return B(3, n)
print([A338709(n) for n in range(1, 11)])

Formula

Empirical g.f.: 3*x*(2 + 15*x - 53*x^2 + 89*x^3 - 37*x^4) / ((1 - x)^2 * (1 - 3*x)^2 * (1 - 6*x + 3*x^2)). - Vaclav Kotesovec, Dec 19 2020