A116903 -id:A116903 - cp's OEIS frontend

A116904 Number of n-step self-avoiding walks on the upper 4 octants of the cubic grid starting at origin.

1, 5, 21, 93, 409, 1853, 8333, 37965, 172265, 787557, 3593465, 16477845, 75481105, 346960613, 1593924045, 7341070889, 33798930541, 155915787353, 719101961769, 3321659652529, 15341586477457, 70944927549085, 328054694768261, 1518490945278377, 7028570356547189, 32560476643826933, 150838831585499069

Offset: 0

Giovanni Resta, Feb 15 2006

nonn

Guttmann-Torrie simple cubic lattice series coefficients c_n^{2}(Pi). - N. J. A. Sloane, Jul 06 2015

			See A116903 for a graphical example of the bidimensional counterpart.

M. N. Barber et al., Some tests of scaling theory for a self-avoiding walk attached to a surface, 1978 J. Phys. A: Math. Gen. 11 1833.
Nathan Clisby, Andrew R. Conway and Anthony J. Guttmann, Three-dimensional terminally attached self-avoiding walks and bridges, J. Phys. A: Math. Theor., 49 (2016), 015004; arXiv:1504.02085 [cond-mat.stat-mech], 2015. [Warning: arXiv version has typos in a(11) and a(12).]
T. Dachraoui et al., Elementary paths in a cubic lattice and application to molecular biology, Kybernetes, Vol. 26 No. 9, pp. 1012-1030.
A. J. Guttmann and G. M. Torrie, Critical behavior at an edge for the SAW and Ising model, J. Phys. A 17 (1984), 3539-3552.

Cf. A001412, A039648, A116903.

a(16)-a(20) from Scott R. Shannon, Aug 12 2020

a(21)-a(26) from Clisby et al. added by Andrey Zabolotskiy, Apr 18 2023

A335780 The number of hanging vertically stable self-avoiding walks of length n on a 2D square lattice where both the nodes and connecting rods have mass.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 7, 15, 37, 65, 115, 223, 503, 1127, 2761, 6225, 15393, 34915, 84399, 193489, 477727, 1113059, 2753799, 6486011, 16181965, 38447093, 95995579

Offset: 1

Scott R. Shannon, Sep 13 2020

Consider a self-avoiding walk on a 2D square lattice where each visited node is given a fixed mass and each node is connected by a rod of another fixed mass. Hang the resulting lattice structure from a string at the first node. This sequence gives the number of walks of length n such that the structure will hang perfectly vertically, and will return to this position if perturbed.
For a walk to be stable requires the torque around the first node to be zero for both the node and rod masses, and that the overall center of mass of the structure is lower than the first node. As n increases the number of walks satisfying these conditions decreases rapidly. For example the total number of 2D self-avoiding walks on a square lattice in the lower two quadrants for n=27 is A116903(27) = 227399388019. The total number of hanging stable walks for n=27 is 95995579, indicating only one in about 2370 walks is stable.
For all stable walks it is found that the final node is always directly underneath the starting node. This is not the case if only the node or rod masses are considered.
See A337761 for the equalivalent sequence on a 3D cubic lattice.

			a(1)-a(5) = 1 as the only stable walk is a walk straight down from the first node.
a(6) = 3. There is one stable walk with a first step to the right:
.
      X-----+
            |
            |
+-----+-----+
|
|
+-----+
.
where 'X' represents the hanging point first node at (0,0).
Assuming a mass of p for the nodes, q for the rods, and a length l for the rods, the total torque from the nodes to the right of the first node is 2*p*l, which equals that from the nodes to the left. The total torque for the rods to the right of the first node is 2*q*(1/2)*l + 1*q*1*l = 2ql, which equals that from the rods to the left. The center of mass is at coordinate (0,-1). This walk can be taken in 2 ways thus, with the straight down walk, the total number of stable walks is 2+1 = 3.
a(20) = 193489. An example of a 20-step stable walk is:
.
            X---+
                |
    +---+       +---+---+
    |   |               |
    +   +---+---+       +
    |           |       |
+---+           +---+---+
|
+---+---+---+
.
The total torque from the nodes to the right of the first node is 4*p*1*l + 2*p*2*l + 3*p*3*l = 17pl. The torque from the left nodes is 3*p*1*l + 4*p*2*l + 2*p*3*l = 17pl. The total torque from the rods to the right of the first node is 2*q*(l/2)*l + 2*q*1*l + 2*q*(3/2)*l + 2*q*(5/2)*l + 2*q*3*l = 17ql. The torque from the rods on the left is 2*q*(l/2)*l + 1*q*1*l + 2*q*(3/2)*l + 2*q*2*l + 2*q*(5/2)*l + 1*q*3*l = 17ql. This shows the configuration does not have to be symmetrical to be balanced.
See the linked text file for the step directions for the stable walks for n=6 to n=15.

Scott R. Shannon, Stable walks for n=6 to n=15.

Cf. A116903, A337761, A001411, A001412.

A336818 Table read by antidiagonals: T(b,n) is the number of n-step self avoiding walks on a 2D square grid confined inside a square box of size 2b X 2b where the walk starts at the middle of the box.

Original entry on oeis.org

4, 8, 4, 8, 12, 4, 8, 32, 12, 4, 8, 64, 36, 12, 4, 8, 104, 96, 36, 12, 4, 8, 176, 240, 100, 36, 12, 4, 8, 296, 520, 280, 100, 36, 12, 4, 0, 496, 1048, 728, 284, 100, 36, 12, 4, 0, 848, 2104, 1816, 776, 184, 100, 36, 12, 4, 0, 1392, 4168, 4176, 2112, 780, 284, 100, 36, 12, 4

Offset: 1

Scott R. Shannon, Aug 06 2020

			T(1,3) = 8. The one 3-step walk taking a first step to the right followed by a step upward is:
.
+--+
   |
*--+
.
This walk can take a downward second step, and also have a first step in the four possible directions, given a total of 1*2*4 = 8 total walks.
.
The table begins:
.
4  8  8   8   8   8    8    8     0     0      0      0      0       0       0...
4 12 32  64 104 176  296  496   848  1392   2280   3624   5472    8200   10920...
4 12 36  96 240 520 1048 2104  4168  8288  16488  32536  64680  126560  248328...
4 12 36 100 280 728 1816 4176  9304 20400  44216  95680 206104  442984  953720...
4 12 36 100 284 776 2112 5448 13704 32824  77232 178552 409144  932152 2113736...
4 12 36 100 284 780 2168 5848 15672 40472 102816 252992 615328 1472808 3501200...
4 12 36 100 284 780 2172 5912 16192 43360 115328 298856 765864 1919328 4770784...
4 12 36 100 284 780 2172 5916 16264 44016 119392 318328 843848 2194920 5664648...
4 12 36 100 284 780 2172 5916 16268 44096 120200 323856 872920 2321600 6146400...
4 12 36 100 284 780 2172 5916 16268 44100 120288 324832 880232 2363520 6344240...
4 12 36 100 284 780 2172 5916 16268 44100 120292 324928 881392 2372968 6402928...
4 12 36 100 284 780 2172 5916 16268 44100 120292 324932 881496 2374328 6414896...
4 12 36 100 284 780 2172 5916 16268 44100 120292 324932 881500 2374440 6416472...
4 12 36 100 284 780 2172 5916 16268 44100 120292 324932 881500 2374444 6416592...
4 12 36 100 284 780 2172 5916 16268 44100 120292 324932 881500 2374444 6416596...
...

A. R. Conway et al., Algebraic techniques for enumerating self-avoiding walks on the square lattice, J. Phys A 26 (1993) 1519-1534.
A. J. Guttmann and A. R. Conway, Self-Avoiding Walks and Polygons, Annals of Combinatorics 5 (2001) 319-345.

Cf. A001411 (b->infinity), A336872 (start on edge of box), A116903, A038373.

For n <= b, T(b,n) = A001411(n).

For n >= b^2, T(b,n) = 0 as the walks have more steps than there are free grid points inside the box.

A337317 The number of stable vertically balanced self-avoiding walks of length n on the upper half-plane of a 2D square lattice where the nodes and connecting rods have equal mass.

Original entry on oeis.org

2, 4, 10, 24, 60, 138, 348, 832, 2104, 5192, 13178, 32662, 82890, 207888, 529738, 1339188, 3424526, 8698382, 22294906, 56836056, 145982928, 373363770, 960834764, 2463930512, 6351046936, 16322104184, 42131167144, 108478565772, 280360764620

Offset: 1

Scott R. Shannon, Sep 28 2020

This is a variation of A337860 where only walks which are stable against a small perturbation from either left or right are counted. This means any walks which have their center-of-mass directly above the extrema of the nodes touching the y=0 starting line are not counted, e.g. a walk directly up from the first node.

See A337860 for further details and examples of the walks in this sequence.

			a(1) = 2. The two stable walks are a single step left or right from the first node. The walk consisting of a single vertical step is not counted, as it has its center-of-mass directly above the single node touching the y=0 line and will thus topple with a slight perturbation from either the left or right directions.
a(3) = 10. The stable 3-step walks with a first step up or to the right are:
.
                                            +
+---+                         +  +---+      |
|   |  X---+---+---+          |      |      +
X   +                 X---+---+  X---+      |
                                        X---+
.
These walks can also be taken with a first or second step to the left, giving a total number of stable walks of 2*5 = 10.
The semi-stable 3-step walks which are not counted in this sequence, but are counted in A337860, are:
.
                        +
                        |
    +---+   +---+       +
    |           |       |
X---+           +---X   +
                        |
                        X
.
as a slight perturbation from the left, right, and left or right would topple the first, second and third structure respectively.

Cf. A337860 (count semi-stable walks), A335780, A337761, A116903, A116904, A001411.

A337860 The number of vertically balanced self-avoiding walks of length n on the upper half-plane of a 2D square lattice where the nodes and connecting rods have equal mass.

Original entry on oeis.org

3, 5, 13, 27, 65, 145, 361, 855, 2163, 5303, 13419, 33195, 84159, 210765, 536871, 1356153, 3466533, 8799247, 22541583, 57428441, 147423495, 376838119, 969292869, 2484478265, 6401330591, 16445203213, 42434086359, 109225591309, 282209330237

Offset: 1

Scott R. Shannon, Sep 27 2020

Consider a self-avoiding walk in the upper half-plane on a 2D square lattice where each visited node is given a fixed mass and each node is connected by a rod of the same mass. Let the resulting lattice structure be free to move in a downward gravitational field. This sequence gives the number of walks of length n such that the structure will remain in place and will not topple given no sideways perturbations.
For a walk to be stable requires the center-of-mass of the resulting structure to be above or inside the extrema of the horizontal positions of the nodes that are on the y=0 line where the walk begins. Here we assume no perturbations so allow walks which would topple if either a left or right perturbation acts, for example we allow a directly vertical walk above the starting node. For the number of walks where such semi-stable structures are not counted see A337317.
We also assume the nodes and the rods are of equal mass. This is required as some structures exist which are either stable or would topple depending on the relative mass of the nodes and rods. For example the 8-step walk:
.
    +---+---+
            |
            +
            |
        +---+
        |
X---+---+
.
Considering only the nodes the center-of-mass is at position 17/9 (~1.88) relative to the starting x=0 'X' position - this is between the x=0 and x=2 extrema of the nodes at y=0 and is thus stable. Considering only the rods the center-of-mass is at position 33/16 (~2.06) relative to 'X' - this is to the right of the node at x=2 and thus the structure would topple to the right. To avoid such issues we assume both rods and nodes are of equal mass. Given that, the center-of-mass of this walk is at 67/34 (~1.97) and is thus stable.
The number of stable walks in this sequence does not decrease as rapidly as compared to the number of hanging 2D stable walks of A335780. For example the total number of 2D self-avoiding walks on a square lattice in the upper half plane for n=29 is A116903(27) = 1577923781445. The total number of vertically stable walks here for n=29 is 282209330237, indicating about 1 in 6 walks are stable. This is expected as many otherwise unstable walks becomes stable if some node touches the y=0 line away from the starting node; this becomes relatively common as n increases. Any of the symmetrical walks in A335780 which have no nodes above the starting node will also be in this sequence, inverted from top to bottom.

			a(3) = 13. The stable 3-step walks with a first step upward or to the right are:
.
                                                              +
                                                          +   |
                        +      +---+   +---+   +---+      |   +
                        |      |           |   |   |      +   |
X---+---+---+   X---+---+  X---+       X---+   X   +      |   +
                                                      X---+   |
                                                              X
.
The first six walks can also be taken with a first or second step to the left, giving a total number of stable walks of 2*6 + 1 = 13. Note that the third walk would topple with a perturbation to the right, and the final walk would topple with a perturbation to either the left or right.
The three non-stable 3-step walks in the first quadrant are:
.
    +               +---+
    |               |
+---+   +---+---+   +
|       |           |
X       X           X
.
These can also be taken with a second step to the left, giving six unstable walks.
a(23) = 969292869. An example of a stable 23-step walk with a base of 1 unit is:
.
                        +---+
                        |   |
    +---+---+---+---+---+   +
    |                       |
+---+               +---+   +
|                   |   |   |
+---+---+---+   +---+   +---+
            |   |
            +   X
.

Cf. A337317 (do not count semi-stable walks), A335780, A337761, A116903, A116904, A001411.

A335307 The number of hanging vertically stable self-avoiding walks of length n on a 2D square lattice where only the nodes have mass.

Original entry on oeis.org

1, 1, 1, 1, 5, 13, 31, 63, 141, 293, 665, 1553, 3795, 9225, 22257, 53623, 132277, 321651, 786553, 1928565, 4806503, 11885969, 29498995, 73362933, 184210629, 460165983, 1151961103

Offset: 1

Scott R. Shannon, Sep 12 2020

This is a variation of A335780 where only the nodes have mass. See that sequence for further details of the allowed walks.

			a(1)-a(4) = 1 as the only stable walk is a walk straight down from the first node.
a(5) = 5. There are two stable walks with a first step to the right:
.
      X-----+
            |     +     X-----+
            |     |           |
+-----+-----+     |           |
|                 +-----+-----+
|
+
.
Assuming a node mass of p, both walks have a torque of 2p to the right and 2p to the left of the first node. These walks can be taken in 2 ways. Thus, with the straight down walk, the total number of stable walks is 2*2+1 = 5.

Cf. A335780 (rods and nodes have mass), A335596 (only rods have mass), A116903, A337761, A001411, A001412.

A335596 The number of hanging vertically stable self-avoiding walks of length n on a 2D square lattice where only the connecting rods have mass.

Original entry on oeis.org

1, 1, 1, 1, 3, 7, 17, 43, 91, 183, 371, 799, 1941, 4621, 11463, 27823, 68997, 167481, 414045, 1006091, 2496981, 6127053, 15304071, 37838777, 95041475, 236320611, 595206771

Offset: 1

Scott R. Shannon, Sep 13 2020

This is a variation of A335780 where only the rods between the nodes have mass. See that sequence for further details of the allowed walks.

			a(1)-a(4) = 1 as the only stable walk is a walk straight down from the first node.
a(5) = 3. There is one stable walk with a first step to the right:
.
            X-----+
                  |
                  |
+-----+-----+-----+
,
Assuming a rod mass of q, the total torque to the right of the first node is 2*q*(1/2) + 1*q*1 = 2q. The total torque to the left of the first node is 1*q*(1/2) + 1*q*(3/2) = 2q. This walk can be taken in 2 ways. Thus, with the straight down walk, the total number of stable walks is 2+1 = 3.

Cf. A335780 (rods and nodes have mass), A335307 (only nodes have mass), A116903, A337761, A001411, A001412.

A336769 Table read by antidiagonals: T(h,n) is the number of n-step self avoiding walks on a 2D square grid confined to an infinite strip of height h where the walk starts at the origin.

Original entry on oeis.org

3, 6, 3, 12, 7, 3, 20, 18, 7, 3, 36, 40, 19, 7, 3, 58, 86, 48, 19, 7, 3, 100, 170, 120, 49, 19, 7, 3, 160, 350, 274, 130, 49, 19, 7, 3, 268, 688, 620, 326, 131, 49, 19, 7, 3, 430, 1394, 1346, 810, 338, 131, 49, 19, 7, 3, 708, 2702, 2972, 1912, 884, 339, 131, 49, 19, 7, 3

Offset: 1

Scott R. Shannon, Aug 04 2020

			T(1,3) = 12. The six 3-step walks taking a first step to the right or a first step upward followed by a step to the right are:
.
                  +  +--+     +--+  +--+--+  +--+
                  |     |     |     |        |  |
+--+--+--+  +--+--+  +--+  +--+     +        +  +
.
The same steps can be taken to the left, giving a total of 2*6 = 12 walks.
.
The table begins:
.
3 6 12 20  36  58 100  160  268   430   708   1140   1860   3002    4876    7880...
3 7 18 40  86 170 350  688 1394  2702  5338  10278  20078  38578   74820  143496...
3 7 19 48 120 274 620 1346 2972  6402 13994  29870  64412 136308  291008  612920...
3 7 19 49 130 326 810 1912 4486 10262 23634  53642 122624 276524  627248 1405154...
3 7 19 49 131 338 884 2228 5560 13438 32320  76440 181202 425138 1001128 2336886...
3 7 19 49 131 339 898 2328 6050 15320 38478  94642 231798 560794 1357098 3258148...
3 7 19 49 131 339 899 2344 6180 16040 41572 105806 267560 666682 1655140 4070280...
3 7 19 49 131 339 899 2345 6198 16204 42586 110636 286682 733032 1865008 4693178...
3 7 19 49 131 339 899 2345 6199 16224 42788 112016 293908 764248 1982070 5089002...
3 7 19 49 131 339 899 2345 6199 16225 42810 112260 295734 774682 2030988 5286652...
3 7 19 49 131 339 899 2345 6199 16225 42811 112284 296024 777042 2045610 5360672...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296050 777382 2048600 5380646...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777410 2048994 5384370...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777411 2049024 5384822...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777411 2049025 5384854...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777411 2049025 5384855...
...

Cf. A116903 (h->infinity), A038577 (h=1), A302408 (h=2), A001411, A038373.

For n <= h, T(h,n) = A116903(n).
Row 1 = T(1,n) = A038577(n).
Row 2 = T(2,n) = A302408(n).

A336863 Table read by antidiagonals: T(b,n) is the number of n-step self avoiding walks on a 2D square grid confined inside an infinite well of width 2b where the walk starts at the middle of the well bottom.

Original entry on oeis.org

3, 5, 3, 11, 7, 3, 19, 17, 7, 3, 41, 39, 19, 7, 3, 79, 85, 47, 19, 7, 3, 163, 187, 119, 49, 19, 7, 3, 163, 187, 119, 49, 19, 7, 3, 305, 425, 273, 129, 49, 19, 7, 3, 603, 955, 657, 325, 131, 49, 19, 7, 3, 1143, 2169, 1517, 809, 337, 131, 49, 19, 7, 3

Offset: 1

Scott R. Shannon, Aug 07 2020

			The infinite well of width 2b is:
.                           .
.                           .
+                           +
|                           |
+                           +
|                           |
+---+-- ... --X-- ... --+---+
<------b----->
.
T(1,3) = 11. The five 3-step walks taking a first step to the right or upward steps followed by a step to the right are:
.
   +                    +   +--+
   |                    |   |
   +   +--+   +--+   +--+   +
   |      |   |  |   |      |
*--+   *--+   *  +   *      *
.
These walks can also take similar steps to the left. There is also one 3-step walk directly upward, given a total of 5*2+1 = 11 walks.
The table begins:
.
3 5 11 19  41  79 163  305  603  1143  2231   4257   8233  15721   30265   57871...
3 7 17 39  85 187 425  955 2169  4867 10961  24439  54583 121079  269073  595295...
3 7 19 47 119 273 657 1517 3645  8517 20435  48029 114961 270681  645759 1519165...
3 7 19 49 129 325 809 1979 4817 11703 28475  69255 168749 410905 1002425 2443189...
3 7 19 49 131 337 883 2227 5669 14017 35109  86465 215531 531041 1321687 3260577...
3 7 19 49 131 339 897 2327 6049 15485 39421  99651 251065 631073 1584165 3973513...
3 7 19 49 131 339 899 2343 6179 16039 41809 107261 276041 701555 1790849 4530571...
3 7 19 49 131 339 899 2345 6197 16203 42585 110963 288833 746717 1925057 4942513...
3 7 19 49 131 339 899 2345 6199 16223 42787 112015 294345 767319 2003283 5188119...
3 7 19 49 131 339 899 2345 6199 16225 42809 112259 295733 775251 2035247 5318433...
3 7 19 49 131 339 899 2345 6199 16225 42811 112283 296023 777041 2046335 5366435...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296049 777381 2048599 5381553...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777409 2048993 5384369...
...

A. R. Conway et al., Algebraic techniques for enumerating self-avoiding walks on the square lattice, J. Phys A 26 (1993) 1519-1534.
A. J. Guttmann and A. R. Conway, Self-Avoiding Walks and Polygons, Annals of Combinatorics 5 (2001) 319-345.

Cf. A116903 (b->infinity), A001411, A038373.

For n <= b, T(b,n) = A116903(n).

A336872 Table read by antidiagonals: T(b,n) is the number of n-step self avoiding walks on a 2D square grid confined inside a square box of dimension 2b X 2b where the walk starts at the middle of one of the box's edges.

Original entry on oeis.org

3, 5, 3, 10, 7, 3, 10, 17, 7, 3, 16, 39, 19, 7, 3, 10, 84, 47, 19, 7, 3, 14, 174, 119, 49, 19, 7, 3, 0, 336, 273, 129, 49, 19, 7, 3, 0, 634, 656, 325, 131, 49, 19, 7, 3, 0, 1072, 1500, 809, 337, 131, 49, 19, 7, 3, 0, 1856, 3496, 1979, 883, 339, 131, 49, 19, 7, 3

Offset: 1

Scott R. Shannon, Aug 06 2020

			T(1,3) = 10. The five 3-step walks taking a first step to the right or upward steps followed by a step to the right are:
.
   +                    +   +--+
   |                    |   |
   +   +--+   +--+   +--+   +
   |      |   |  |   |      |
*--+   *--+   *  +   *      *
.
This walk can also take similar steps to the left, given a total of 5*2 = 10 walks.
.
The table begins:
.
3 5 10 10  16  10  14    0    0     0     0      0      0      0       0       0...
3 7 17 39  84 174 336  634 1072  1856  2888   4598   6526   9198   11504   13758...
3 7 19 47 119 273 656 1500 3496  7612 16762  34214  71932 140664  286522  540490...
3 7 19 49 129 325 809 1979 4816 11682 28250  67606 159380 370530  842432 1902126...
3 7 19 49 131 337 883 2227 5669 14017 35108  86440 215214 528312 1303650 3162374...
3 7 19 49 131 339 897 2327 6049 15485 39421  99651 251064 631044 1583740 3969304...
3 7 19 49 131 339 899 2343 6179 16039 41809 107261 276041 701555 1790848 4530538...
3 7 19 49 131 339 899 2345 6197 16203 42585 110963 288833 746717 1925057 4942513...
3 7 19 49 131 339 899 2345 6199 16223 42787 112015 294345 767319 2003283 5188119...
3 7 19 49 131 339 899 2345 6199 16225 42809 112259 295733 775251 2035247 5318433...
3 7 19 49 131 339 899 2345 6199 16225 42811 112283 296023 777041 2046335 5366435...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296049 777381 2048599 5381553...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777409 2048993 5384369...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777411 2049023 5384821...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777411 2049025 5384853...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777411 2049025 5384855...
...

A. R. Conway et al., Algebraic techniques for enumerating self-avoiding walks on the square lattice, J. Phys A 26 (1993) 1519-1534.
A. J. Guttmann and A. R. Conway, Self-Avoiding Walks and Polygons, Annals of Combinatorics 5 (2001) 319-345.

Cf. A116903 (b->infinity), A336818 (start at middle of box), A001411, A038373.

For n <= b, T(b,n) = A116903(n).

For n >= b^2, T(b,n) = 0 as the walks have more steps than there are free grid points inside the box.

cp's OEIS Frontend

A116904 Number of n-step self-avoiding walks on the upper 4 octants of the cubic grid starting at origin.

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A335780 The number of hanging vertically stable self-avoiding walks of length n on a 2D square lattice where both the nodes and connecting rods have mass.

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A336818 Table read by antidiagonals: T(b,n) is the number of n-step self avoiding walks on a 2D square grid confined inside a square box of size 2b X 2b where the walk starts at the middle of the box.

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A337317 The number of stable vertically balanced self-avoiding walks of length n on the upper half-plane of a 2D square lattice where the nodes and connecting rods have equal mass.

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A337860 The number of vertically balanced self-avoiding walks of length n on the upper half-plane of a 2D square lattice where the nodes and connecting rods have equal mass.

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A335307 The number of hanging vertically stable self-avoiding walks of length n on a 2D square lattice where only the nodes have mass.

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A335596 The number of hanging vertically stable self-avoiding walks of length n on a 2D square lattice where only the connecting rods have mass.

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A336769 Table read by antidiagonals: T(h,n) is the number of n-step self avoiding walks on a 2D square grid confined to an infinite strip of height h where the walk starts at the origin.

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A336863 Table read by antidiagonals: T(b,n) is the number of n-step self avoiding walks on a 2D square grid confined inside an infinite well of width 2b where the walk starts at the middle of the well bottom.

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A336872 Table read by antidiagonals: T(b,n) is the number of n-step self avoiding walks on a 2D square grid confined inside a square box of dimension 2b X 2b where the walk starts at the middle of one of the box's edges.

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