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)
A333323
Number of self-avoiding closed paths on an n X n grid which pass through NW and SE corners.
Original entry on oeis.org
1, 3, 42, 1799, 232094, 92617031, 115156685746, 442641690778179, 5224287477491915786, 188825256606226776728029, 20879416139356164466643759334, 7057757437924198729598570424130207, 7287699030020917172151307665469211016474, 22973720258279267139936821063450448822110219653
Offset: 2
a(2) = 1;
+--*
| |
*--+
a(3) = 3;
+--*--* +--*--* +--*
| | | | | |
*--* * * * * *--*
| | | | | |
*--+ *--*--+ *--*--+
- Anthony J. Guttmann and Iwan Jensen, Table of n, a(n) for n = 2..27
- Anthony J. Guttmann and Iwan Jensen, Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices, arXiv:2208.06744 [math-ph], Aug 13 2022, Table D2 (with offset 1).
- Anthony J. Guttmann and Iwan Jensen, The gerrymander sequence, or A348456, arXiv:2211.14482 [math.CO], 2022.
-
# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333323(n):
universe = tl.grid(n - 1, n - 1)
GraphSet.set_universe(universe)
cycles = GraphSet.cycles().including(1).including(n * n)
return cycles.len()
print([A333323(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)])
A333651
Triangle T(n,k), n >= 2, 0 <= k <= floor(n^2/2)-2, read by rows, where T(n,k) is the number of 2*(k+2)-cycles in the n X n grid graph which pass through NW corner (0,0).
Original entry on oeis.org
1, 1, 2, 4, 1, 2, 6, 18, 40, 24, 6, 1, 2, 6, 20, 72, 248, 698, 1100, 1096, 662, 206, 1, 2, 6, 20, 74, 298, 1228, 4762, 15984, 40026, 75524, 109150, 121130, 99032, 51964, 11996, 1072, 1, 2, 6, 20, 74, 300, 1300, 5844, 26148, 110942, 427388, 1393796, 3790524, 8648638, 16727776, 27529284, 38120312, 43012614, 37385280, 23166526, 9496426, 2286972, 242764
Offset: 2
T(3,0) = 1;
+--*
| |
*--*
T(3,1) = 2;
+--*--* +--*
| | | |
*--*--* * *
| |
*--*
T(3,2) = 4;
+--*--* +--*--* +--*--* +--*
| | | | | | | |
* * * *--* *--* * * *--*
| | | | | | | |
*--*--* *--* *--* *--*--*
Triangle starts:
===================================================
n\k| 0 1 2 3 4 5 6 ... 10 ... 16
---|-----------------------------------------------
2 | 1;
3 | 1, 2, 4;
4 | 1, 2, 6, 18, 40, 24, 6;
5 | 1, 2, 6, 20, 72, 248, 698, ... , 206;
6 | 1, 2, 6, 20, 74, 298, 1228, .......... , 1072;
7 | 1, 2, 6, 20, 74, 300, 1300, ...
8 | 1, 2, 6, 20, 74, 300, 1302, ...
9 | 1, 2, 6, 20, 74, 300, 1302, ...
-
# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333651(n):
universe = tl.grid(n - 1, n - 1)
GraphSet.set_universe(universe)
cycles = GraphSet.cycles().including(1)
return [cycles.len(2 * k).len() for k in range(2, n * n // 2 + 1)]
print([i for n in range(2, 8) for i in A333651(n)])
Original entry on oeis.org
2, 8, 98, 4112, 532270, 212372938, 263708907212, 1013068026356376, 11955420069208095720, 432101605951906251627394, 47778407166747833830058004150, 16149888968763663448192636077980754, 16675786862526496319891707194153887550752, 52568166380872328447478940416604864445574575710
Offset: 2
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