A121785
"Spanning walks" on the square lattice (see Jensen web site for further information).
Original entry on oeis.org
8, 95, 2320, 154259, 30549774, 17777600753, 30283708455564, 152480475641255213, 2287842813828061810244, 102744826737618542833764649, 13848270995235582268846758977770
Offset: 1
A333466
Number of self-avoiding closed paths on an n X n grid which pass through four corners ((0,0), (0,n-1), (n-1,n-1), (n-1,0)).
Original entry on oeis.org
1, 1, 11, 373, 44930, 17720400, 22013629316, 84579095455492
Offset: 2
a(2) = 1;
+--+
| |
+--+
a(3) = 1;
+--*--+
| |
* *
| |
+--*--+
a(4) = 11;
+--*--*--+ +--*--*--+ +--*--*--+
| | | | | |
*--*--* * *--* *--* *--* *
| | | | | |
*--*--* * *--* *--* *--* *
| | | | | |
+--*--*--+ +--*--*--+ +--*--*--+
+--*--*--+ +--*--*--+ +--*--*--+
| | | | | |
* *--*--* * *--* * * *--*
| | | | | | | |
* *--*--* * * * * * *--*
| | | | | | | |
+--*--*--+ +--* *--+ +--*--*--+
+--*--*--+ +--*--*--+ +--* *--+
| | | | | | | |
* * * * * *--* *
| | | | | |
* *--* * * * * *--* *
| | | | | | | | | |
+--* *--+ +--*--*--+ +--* *--+
+--* *--+ +--* *--+
| | | | | | | |
* *--* * * * * *
| | | | | |
* * * *--* *
| | | |
+--*--*--+ +--*--*--+
-
# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333466(n):
universe = tl.grid(n - 1, n - 1)
GraphSet.set_universe(universe)
cycles = GraphSet.cycles()
for i in [1, n, n * (n - 1) + 1, n * n]:
cycles = cycles.including(i)
return cycles.len()
print([A333466(n) for n in range(2, 10)])
-
def search(x, y, n, used)
return 0 if x < 0 || n <= x || y < 0 || n <= y || used[x + y * n]
return 1 if x == 0 && y == 1 && [n - 1, n * (n - 1), n * n - 1].all?{|i| used[i] == true}
cnt = 0
used[x + y * n] = true
@move.each{|mo|
cnt += search(x + mo[0], y + mo[1], n, used)
}
used[x + y * n] = false
cnt
end
def A(n)
return 1 if n < 3
@move = [[1, 0], [-1, 0], [0, 1], [0, -1]]
used = Array.new(n * n, false)
search(0, 0, n, used)
end
def A333466(n)
(2..n).map{|i| A(i)}
end
p A333466(6)
A333246
Number of self-avoiding closed paths on an n X n grid which pass through NW corner.
Original entry on oeis.org
1, 7, 97, 4111, 532269, 212372937, 263708907211, 1013068026356375, 11955420069208095719, 432101605951906251627393, 47778407166747833830058004149, 16149888968763663448192636077980753, 16675786862526496319891707194153887550751, 52568166380872328447478940416604864445574575709
Offset: 2
a(2) = 1;
+--*
| |
*--*
a(3) = 7;
+--* +--*--* +--*--* +--*
| | | | | | | |
*--* *--*--* * * * *
| | | |
*--*--* *--*
+--*--* +--*--* +--*
| | | | | |
* *--* *--* * * *--*
| | | | | |
*--* *--* *--*--*
-
# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333246(n):
universe = tl.grid(n - 1, n - 1)
GraphSet.set_universe(universe)
cycles = GraphSet.cycles().including(1)
return cycles.len()
print([A333246(n) for n in range(2, 10)])
A333247
Number of self-avoiding closed paths on an n X n grid which pass through NW and SW corners.
Original entry on oeis.org
1, 4, 47, 1843, 232905, 92729439, 115234959344, 442748883422394
Offset: 2
a(2) = 1;
+--*
| |
+--*
a(3) = 4;
+--*--* +--*--* +--* +--*
| | | | | | | |
* * * *--* * *--* * *
| | | | | | | |
+--*--* +--* +--*--* +--*
-
# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333247(n):
universe = tl.grid(n - 1, n - 1)
GraphSet.set_universe(universe)
cycles = GraphSet.cycles().including(1).including(n)
return cycles.len()
print([A333247(n) for n in range(2, 10)])
A333667
Triangle T(n,k), n >= 2, 0 <= k <= floor(n^2/2)-2*n+2, read by rows, where T(n,k) is the number of 2*(k+2*n-2)-cycles in the n X n grid graph which pass through NW and SE corners ((0,0),(n-1,n-1)).
Original entry on oeis.org
1, 3, 20, 16, 6, 175, 420, 562, 456, 186, 1764, 8064, 21224, 39500, 55376, 57248, 37586, 10260, 1072, 19404, 138600, 569768, 1717152, 4151965, 8371428, 14126846, 19364732, 20241450, 14759356, 6998166, 1927724, 230440
Offset: 2
T(3,0) = 3;
+--*--* +--*--* +--*
| | | | | |
*--* * * * * *--*
| | | | | |
*--+ *--*--+ *--*--+
Triangle starts:
=======================================================================
n\k| 0 1 2 ... 4 ... 8 ... 12 ... 18
---|-------------------------------------------------------------------
2 | 1;
3 | 3;
4 | 20, 16, 6;
5 | 175, 420, 562, ... , 186;
6 | 1764, 8064, 21224, .......... , 1072;
7 | 19404, 138600, 569768, .................. , 230440;
8 | 226512, 2265120, 12922446, ............................ , 4638576;
-
# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333667(n):
universe = tl.grid(n - 1, n - 1)
GraphSet.set_universe(universe)
cycles = GraphSet.cycles().including(1).including(n * n)
return [cycles.len(2 * k).len() for k in range(2 * n - 2, n * n // 2 + 1)]
print([i for n in range(2, 8) for i in A333667(n)])
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