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.

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

1,1

Links

Table of n, a(n) for n=1..62.

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.

Formula

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

Example

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...
...

Crossrefs

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

Sequence in context: A167221 A177930 A242034 * A205708 A309392 A089730

Adjacent sequences: A336860 A336861 A336862 * A336864 A336865 A336866

Keyword

nonn,walk,tabl

Author

Scott R. Shannon, Aug 07 2020

Status

approved