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%I #28 Feb 21 2021 02:08:54
%S 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,
%T 36,12,4,8,296,520,280,100,36,12,4,0,496,1048,728,284,100,36,12,4,0,
%U 848,2104,1816,776,184,100,36,12,4,0,1392,4168,4176,2112,780,284,100,36,12,4
%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 size 2b X 2b where the walk starts at the middle of the box.
%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) = A001411(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) = 8. The one 3-step walk taking a first step to the right followed by a step upward is:
%e .
%e +--+
%e |
%e *--+
%e .
%e 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.
%e .
%e The table begins:
%e .
%e 4 8 8 8 8 8 8 8 0 0 0 0 0 0 0...
%e 4 12 32 64 104 176 296 496 848 1392 2280 3624 5472 8200 10920...
%e 4 12 36 96 240 520 1048 2104 4168 8288 16488 32536 64680 126560 248328...
%e 4 12 36 100 280 728 1816 4176 9304 20400 44216 95680 206104 442984 953720...
%e 4 12 36 100 284 776 2112 5448 13704 32824 77232 178552 409144 932152 2113736...
%e 4 12 36 100 284 780 2168 5848 15672 40472 102816 252992 615328 1472808 3501200...
%e 4 12 36 100 284 780 2172 5912 16192 43360 115328 298856 765864 1919328 4770784...
%e 4 12 36 100 284 780 2172 5916 16264 44016 119392 318328 843848 2194920 5664648...
%e 4 12 36 100 284 780 2172 5916 16268 44096 120200 323856 872920 2321600 6146400...
%e 4 12 36 100 284 780 2172 5916 16268 44100 120288 324832 880232 2363520 6344240...
%e 4 12 36 100 284 780 2172 5916 16268 44100 120292 324928 881392 2372968 6402928...
%e 4 12 36 100 284 780 2172 5916 16268 44100 120292 324932 881496 2374328 6414896...
%e 4 12 36 100 284 780 2172 5916 16268 44100 120292 324932 881500 2374440 6416472...
%e 4 12 36 100 284 780 2172 5916 16268 44100 120292 324932 881500 2374444 6416592...
%e 4 12 36 100 284 780 2172 5916 16268 44100 120292 324932 881500 2374444 6416596...
%e ...
%Y Cf. A001411 (b->infinity), A336872 (start on edge of box), A116903, A038373.
%K nonn,walk,tabl
%O 1,1
%A _Scott R. Shannon_, Aug 06 2020