OFFSET
1,2
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..500 (terms 1..50 from Seiichi Manyama)
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 (6,-8,-8,16).
FORMULA
Empirical g.f.: x*(1 + 6*x - 16*x^2 + 24*x^3 - 16*x^4) / ((1 - 2*x)^2 * (1 - 2*x - 4*x^2)). - Vaclav Kotesovec, Dec 16 2020
The above formula is correct. - 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 A339760(n):
return B(n, 2)
print([A339760(n) for n in range(1, 21)])
(PARI) Vec((1 + 6*x - 16*x^2 + 24*x^3 - 16*x^4) / ((1 - 2*x)^2 * (1 - 2*x - 4*x^2)) + O(x^20)) \\ Andrew Howroyd, Jan 17 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Dec 16 2020
STATUS
approved