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%I #27 Feb 21 2021 02:09:56
%S 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,
%T 19,7,3,163,187,119,49,19,7,3,305,425,273,129,49,19,7,3,603,955,657,
%U 325,131,49,19,7,3,1143,2169,1517,809,337,131,49,19,7,3
%N 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.
%H A. R. Conway et al., <a href="http://dx.doi.org/10.1088/0305-4470/26/7/012">Algebraic techniques for enumerating self-avoiding walks on the square lattice</a>, J. Phys A 26 (1993) 1519-1534.
%H A. J. Guttmann and A. R. Conway, <a href="http://dx.doi.org/10.1007/PL00013842">Self-Avoiding Walks and Polygons</a>, Annals of Combinatorics 5 (2001) 319-345.
%F For n <= b, T(b,n) = A116903(n).
%e The infinite well of width 2b is:
%e . .
%e . .
%e + +
%e | |
%e + +
%e | |
%e +---+-- ... --X-- ... --+---+
%e <------b----->
%e .
%e 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:
%e .
%e + + +--+
%e | | |
%e + +--+ +--+ +--+ +
%e | | | | | |
%e *--+ *--+ * + * *
%e .
%e 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.
%e The table begins:
%e .
%e 3 5 11 19 41 79 163 305 603 1143 2231 4257 8233 15721 30265 57871...
%e 3 7 17 39 85 187 425 955 2169 4867 10961 24439 54583 121079 269073 595295...
%e 3 7 19 47 119 273 657 1517 3645 8517 20435 48029 114961 270681 645759 1519165...
%e 3 7 19 49 129 325 809 1979 4817 11703 28475 69255 168749 410905 1002425 2443189...
%e 3 7 19 49 131 337 883 2227 5669 14017 35109 86465 215531 531041 1321687 3260577...
%e 3 7 19 49 131 339 897 2327 6049 15485 39421 99651 251065 631073 1584165 3973513...
%e 3 7 19 49 131 339 899 2343 6179 16039 41809 107261 276041 701555 1790849 4530571...
%e 3 7 19 49 131 339 899 2345 6197 16203 42585 110963 288833 746717 1925057 4942513...
%e 3 7 19 49 131 339 899 2345 6199 16223 42787 112015 294345 767319 2003283 5188119...
%e 3 7 19 49 131 339 899 2345 6199 16225 42809 112259 295733 775251 2035247 5318433...
%e 3 7 19 49 131 339 899 2345 6199 16225 42811 112283 296023 777041 2046335 5366435...
%e 3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296049 777381 2048599 5381553...
%e 3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777409 2048993 5384369...
%e ...
%Y Cf. A116903 (b->infinity), A001411, A038373.
%K nonn,walk,tabl
%O 1,1
%A _Scott R. Shannon_, Aug 07 2020