A333513
Square array T(n,k), n >= 2, k >= 2, read by antidiagonals, where T(n,k) is the number of self-avoiding closed paths on an n X k grid which pass through four corners ((0,0), (0,k-1), (n-1,k-1), (n-1,0)).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 7, 11, 7, 1, 1, 17, 49, 49, 17, 1, 1, 41, 229, 373, 229, 41, 1, 1, 99, 1081, 3105, 3105, 1081, 99, 1, 1, 239, 5123, 26515, 44930, 26515, 5123, 239, 1, 1, 577, 24323, 227441, 674292, 674292, 227441, 24323, 577, 1
Offset: 2
Square array T(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 1, 3, 7, 17, 41, ...
1, 3, 11, 49, 229, 1081, ...
1, 7, 49, 373, 3105, 26515, ...
1, 17, 229, 3105, 44930, 674292, ...
1, 41, 1081, 26515, 674292, 17720400, ...
-
# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333513(n, k):
universe = tl.grid(n - 1, k - 1)
GraphSet.set_universe(universe)
cycles = GraphSet.cycles()
for i in [1, k, k * (n - 1) + 1, k * n]:
cycles = cycles.including(i)
return cycles.len()
print([A333513(j + 2, i - j + 2) for i in range(11 - 1) for j in range(i + 1)])
A358698
Number of self-avoiding closed paths in the 7 X n grid graph which pass through all vertices on four (left, right, upper, lower) sides of the graph.
Original entry on oeis.org
1, 21, 408, 6410, 113748, 2002405, 35669433, 633099244, 11240647480, 199480271184, 3540336868535, 62831861216325, 1115122033297714, 19790829247392636, 351241699540793996, 6233729269914805533, 110634310753645173365, 1963503651093439655818, 34847658208568166865562, 618465506517313482341986
Offset: 2
A333759
Number of self-avoiding closed paths in the n X n grid graph which pass through all vertices on four (left, right, upper, lower) sides of the graph.
Original entry on oeis.org
1, 1, 11, 191, 11346, 2002405, 1112939654, 1878223479450
Offset: 2
a(2) = 1;
+--+
| |
+--+
a(3) = 1;
+--+--+
| |
+ +
| |
+--+--+
a(4) = 11;
+--+--+--+ +--+--+--+ +--+--+--+
| | | | | |
+--*--* + +--* *--+ +--* +
| | | | | |
+--*--* + +--* *--+ +--* +
| | | | | |
+--+--+--+ +--+--+--+ +--+--+--+
+--+--+--+ +--+--+--+ +--+--+--+
| | | | | |
+ *--*--+ + *--* + + *--+
| | | | | | | |
+ *--*--+ + * * + + *--+
| | | | | | | |
+--+--+--+ +--+ +--+ +--+--+--+
+--+--+--+ +--+--+--+ +--+ +--+
| | | | | | | |
+ + + + + *--* +
| | | | | |
+ *--* + + + + *--* +
| | | | | | | | | |
+--+ +--+ +--+--+--+ +--+ +--+
+--+ +--+ +--+ +--+
| | | | | | | |
+ *--* + + * * +
| | | | | |
+ + + *--* +
| | | |
+--+--+--+ +--+--+--+
-
# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333759(n):
universe = tl.grid(n - 1, n - 1)
GraphSet.set_universe(universe)
cycles = GraphSet.cycles()
points = [i for i in range(1, n * n + 1) if i % n < 2 or ((i - 1) // n + 1) % n < 2]
for i in points:
cycles = cycles.including(i)
return cycles.len()
print([A333759(n) for n in range(2, 10)])
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.
Original entry on oeis.org
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
a(2) = 1;
+--+
| |
+ +
| |
+ +
| |
+--+
a(3) = 3;
+--+--+ +--+--+ +--+--+
| | | | | |
+--* + + *--+ + +
| | | | | |
+--* + + *--+ + +
| | | | | |
+--+--+ +--+--+ +--+--+
-
N=40; x='x+O('x^N); Vec(x^2/(1-3*x-2*x^2+3*x^3-x^4))
-
# 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)])
A358696
Number of self-avoiding closed paths in the 5 X n grid graph which pass through all vertices on four (left, right, upper, lower) sides of the graph.
Original entry on oeis.org
1, 5, 36, 191, 1123, 6410, 37165, 214515, 1240200, 7165033, 41403125, 239227616, 1382302375, 7987125379, 46150853892, 266666446637, 1540838849619, 8903196975232, 51444004997119, 297251155267189, 1717561649837610, 9924328164015589, 57344252900906673, 331343672343272500, 1914553310297505893, 11062575457823993391, 63921216037276901284
Offset: 2
A358697
Number of self-avoiding closed paths in the 6 X n grid graph which pass through all vertices on four (left, right, upper, lower) sides of the graph.
Original entry on oeis.org
1, 11, 122, 1123, 11346, 113748, 1153742, 11674245, 118180383, 1195822385, 12100751361, 122447319062
Offset: 2
Showing 1-6 of 6 results.
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