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A333760
Number of self-avoiding closed paths in the 4 X n grid graph which pass through all vertices on four (left, right, upper, lower) sides of the graph.
2
1, 3, 11, 36, 122, 408, 1371, 4599, 15437, 51804, 173860, 583476, 1958173, 6571695, 22054863, 74016936, 248403622, 833651844, 2797766831, 9389410251, 31511212505, 105752809368, 354910389192, 1191092559048, 3997351239929, 13415260479675, 45022116630931
OFFSET
2,2
FORMULA
G.f.: x^2/(1-3*x-2*x^2+3*x^3-x^4).
a(n) = 3*a(n-1) + 2*a(n-2) - 3*(a-3) + a(n-4) for n > 5.
EXAMPLE
a(2) = 1;
+--+
| |
+ +
| |
+ +
| |
+--+
a(3) = 3;
+--+--+ +--+--+ +--+--+
| | | | | |
+--* + + *--+ + +
| | | | | |
+--* + + *--+ + +
| | | | | |
+--+--+ +--+--+ +--+--+
PROG
(PARI) N=40; x='x+O('x^N); Vec(x^2/(1-3*x-2*x^2+3*x^3-x^4))
(Python)
# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333758(n, k):
universe = tl.grid(n - 1, k - 1)
GraphSet.set_universe(universe)
cycles = GraphSet.cycles()
points = [i for i in range(1, k * n + 1) if i % k < 2 or ((i - 1) // k + 1) % n < 2]
for i in points:
cycles = cycles.including(i)
return cycles.len()
def A333760(n):
return A333758(4, n)
print([A333760(n) for n in range(2, 15)])
CROSSREFS
Row 4 of A333758.
Sequence in context: A347829 A017937 A017938 * A100068 A260745 A119213
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Apr 04 2020
STATUS
approved