A335305
Number of 2-dimensional closed-loop self-avoiding paths on a square lattice where each path consists of steps of length 1 to n which can be taken in any order.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 16128, 287232, 0, 0, 1843367680, 45589291776, 0, 0
Offset: 1
a(1) to a(6) = 0 as no closed loop paths are possible.
a(7) = 16128. Given the first step has length 1 and is to the right, with the next non-right step being upward, there are 84 different loops. Each of these can be walked in at least 2 ways, with the single perfect square having 48 different possible walks. Each of these in turn can be started with a first step of length 1 to n, and each of these can then be walked in 8 different ways on a 2D square grid. This gives a total number of 7-step paths of 16128. This should be compared with A334720 where for n=7 only 8 paths are possible. See the attached link text file for more details of n=7.
A006782
Number of polygons of length 4n on L-lattice.
Original entry on oeis.org
1, 0, 1, 2, 9, 36, 154, 684, 3128, 14666, 70258, 342766, 1698625, 8532410, 43368153, 222729492, 1154455161, 6033034032, 31760434883, 168311948410, 897327111964, 4810159375532, 25914118276362
Offset: 1
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A056625
a(n) is the total area of all self-avoiding polygons of length 2n on the square lattice.
Original entry on oeis.org
1, 4, 22, 124, 726, 4352, 26614, 165204, 1037672, 6580424, 42062040, 270661328, 1751614248, 11391756176, 74406502814, 487838450116, 3209229661682, 21175301453040, 140097533633112, 929160187609096
Offset: 2
A056634
Number of polygons of length 2n with one hole on square lattice (not allowing rotations).
Original entry on oeis.org
1, 24, 342, 3804, 36802, 326540, 2735308, 22006264, 171912401, 1313614776, 9868281330, 73150450248, 536495354621, 3900954906412, 28164962681150, 202166363020920, 1444068530799525, 10272599729628876
Offset: 8
A302335
Constant coefficient of the quadratic polynomials giving the numbers of 2k-cycles in the n X n grid graph for n >= k-1.
Original entry on oeis.org
0, 1, 4, 26, 164, 1046, 6672, 42790, 275888, 1787624, 11634704
Offset: 1
Let p(k,n) be the number of 2k-cycles in the n X n grid graph for n >= k-1. p(k,n) are quadratic polynomials in n, with the first few given by:
p(1,n) = 0,
p(2,n) = 1 - 2*n + n^2,
p(3,n) = 4 - 6*n + 2*n^2,
p(4,n) = 26 - 28*n + 7*n^2,
p(5,n) = 164 - 140*n + 28*n^2,
p(6,n) = 1046 - 740*n + 124*n^2,
p(7,n) = 6672 - 4056*n + 588*n^2,
p(8,n) = 42790 - 22904*n + 2938*n^2,
p(9,n) = 275888 - 132344*n + 15268*n^2,
...
The constant coefficients give a(n), so the first few are 0, 1, 4, 26, 164, .... - _Eric W. Weisstein_, Apr 05 2018
Cf.
A002931 (quadratic coefficients).
A336742
Number of self-avoiding cycles of length 2n on the half-Manhattan lattice.
Original entry on oeis.org
1, 0, 1, 1, 2, 6, 17, 49, 156, 520, 1762, 6110, 21713, 78469, 287393, 1065951, 3998286, 15140500, 57817817, 222478829, 861952579, 3360043843, 13171307962
Offset: 0
A006772
Sum of spans of 2n-step polygons on square lattice.
Original entry on oeis.org
0, 1, 3, 14, 70, 370, 2028, 11452, 66172, 389416, 2326202, 14070268, 86010680, 530576780, 3298906810, 20653559846, 130099026600, 823979294284, 5244162058026, 33523117491920, 215150177410088, 1385839069134800, 8956173544332434, 58056703069399056, 377396656568011618, 2459614847765495754, 16068572108927106202
Offset: 1
From _Andrey Zabolotskiy_, Nov 09 2018: (Start)
There are no 2-step polygons (conventionally).
For n=2, the only 4-step polygon is a 1 X 1 square having span 1, so a(2)=1.
For n=3, the only 6-step polygon is a 2 X 1 domino which can be rotated 2 ways having spans 2 and 1, so a(3) = 2+1 = 3.
For n=4, there are the following 8-step polygons:
a 3 X 1 stick which can be rotated 2 ways having spans 3 and 1;
an L-tromino which can be rotated 4 ways, all having span 2;
a 2 X 2 square, having span 2.
So a(4) = 3 + 1 + 4*2 + 2 = 14.
For n=5, there are the following 10-step polygons:
a 4 X 1 stick which can be rotated 2 ways having spans 4 and 1;
an L-tetromino which can be rotated 2 ways with span 2 and 2 more ways with span 3, plus reflections;
a T-tetromino which can be rotated 2 ways with span 2 and 2 more ways with span 3;
an S-tetromino which can be rotated 2 ways having spans 3 and 2, plus reflections;
a 3 X 2 rectangle which can be rotated 2 ways having spans 3 and 2;
a 3 X 2 rectangle without one of its angular squares having same counts as L-tetromino.
So a(5) = 4 + 1 + 2 * 2*2*(2+3) + 2*(2+3) + 2*(3+2) + 3 + 2 = 70.
(End)
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A056631
a(n) is the total second area moment of all self-avoiding polygons of length 2n on the square lattice.
Original entry on oeis.org
1, 8, 70, 560, 4358, 33160, 248998, 1851040, 13655432, 100126648, 730548788, 5308524968, 38442000664, 277565593032, 1999068564026, 14365917755936, 103038218758426, 737765745264544, 5274413814993896, 37655943519835560
Offset: 2
A056632
a(n) is the total perimeter of all self-avoiding polygons of area n on the square lattice.
Original entry on oeis.org
4, 12, 48, 188, 740, 2936, 11672, 46388, 184352, 732608, 2911040, 11565964, 45949600, 182538264, 725108640, 2880270032, 11440564516, 45441048384, 180483666608, 716832688844, 2847012624636, 11307166941460
Offset: 1
A056638
Number of polygons of length 2n with two (self-avoiding polygon) holes on square lattice (not allowing rotations).
Original entry on oeis.org
2, 122, 3400, 64210, 959742, 12294090, 141329502, 1500596864, 14998956452, 143004562082, 1312929589140, 11688960621918, 101451353794222, 861927965928610, 7191518458613614, 59078731187602024
Offset: 12
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