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%I #28 Feb 21 2021 02:10:00
%S 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,
%T 7,3,0,336,273,129,49,19,7,3,0,634,656,325,131,49,19,7,3,0,1072,1500,
%U 809,337,131,49,19,7,3,0,1856,3496,1979,883,339,131,49,19,7,3
%N Table read by antidiagonals: T(b,n) is the number of n-step self avoiding walks on a 2D square grid confined inside a square box of dimension 2b X 2b where the walk starts at the middle of one of the box's edges.
%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).
%F For n >= b^2, T(b,n) = 0 as the walks have more steps than there are free grid points inside the box.
%e 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:
%e .
%e + + +--+
%e | | |
%e + +--+ +--+ +--+ +
%e | | | | | |
%e *--+ *--+ * + * *
%e .
%e This walk can also take similar steps to the left, given a total of 5*2 = 10 walks.
%e .
%e The table begins:
%e .
%e 3 5 10 10 16 10 14 0 0 0 0 0 0 0 0 0...
%e 3 7 17 39 84 174 336 634 1072 1856 2888 4598 6526 9198 11504 13758...
%e 3 7 19 47 119 273 656 1500 3496 7612 16762 34214 71932 140664 286522 540490...
%e 3 7 19 49 129 325 809 1979 4816 11682 28250 67606 159380 370530 842432 1902126...
%e 3 7 19 49 131 337 883 2227 5669 14017 35108 86440 215214 528312 1303650 3162374...
%e 3 7 19 49 131 339 897 2327 6049 15485 39421 99651 251064 631044 1583740 3969304...
%e 3 7 19 49 131 339 899 2343 6179 16039 41809 107261 276041 701555 1790848 4530538...
%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 3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777411 2049023 5384821...
%e 3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777411 2049025 5384853...
%e 3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777411 2049025 5384855...
%e ...
%Y Cf. A116903 (b->infinity), A336818 (start at middle of box), A001411, A038373.
%K nonn,walk,tabl
%O 1,1
%A _Scott R. Shannon_, Aug 06 2020
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