OFFSET
1,1
LINKS
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
KEYWORD
AUTHOR
Scott R. Shannon, Aug 07 2020
STATUS
approved