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Table read by antidiagonals: T(h,n) is the number of n-step self avoiding walks on a 2D square grid confined to an infinite strip of height h where the walk starts at the origin.
1

%I #26 Feb 01 2021 00:16:31

%S 3,6,3,12,7,3,20,18,7,3,36,40,19,7,3,58,86,48,19,7,3,100,170,120,49,

%T 19,7,3,160,350,274,130,49,19,7,3,268,688,620,326,131,49,19,7,3,430,

%U 1394,1346,810,338,131,49,19,7,3,708,2702,2972,1912,884,339,131,49,19,7,3

%N Table read by antidiagonals: T(h,n) is the number of n-step self avoiding walks on a 2D square grid confined to an infinite strip of height h where the walk starts at the origin.

%F For n <= h, T(h,n) = A116903(n).

%F Row 1 = T(1,n) = A038577(n).

%F Row 2 = T(2,n) = A302408(n).

%e T(1,3) = 12. The six 3-step walks taking a first step to the right or a first step upward followed by a step to the right are:

%e .

%e + +--+ +--+ +--+--+ +--+

%e | | | | | |

%e +--+--+--+ +--+--+ +--+ +--+ + + +

%e .

%e The same steps can be taken to the left, giving a total of 2*6 = 12 walks.

%e .

%e The table begins:

%e .

%e 3 6 12 20 36 58 100 160 268 430 708 1140 1860 3002 4876 7880...

%e 3 7 18 40 86 170 350 688 1394 2702 5338 10278 20078 38578 74820 143496...

%e 3 7 19 48 120 274 620 1346 2972 6402 13994 29870 64412 136308 291008 612920...

%e 3 7 19 49 130 326 810 1912 4486 10262 23634 53642 122624 276524 627248 1405154...

%e 3 7 19 49 131 338 884 2228 5560 13438 32320 76440 181202 425138 1001128 2336886...

%e 3 7 19 49 131 339 898 2328 6050 15320 38478 94642 231798 560794 1357098 3258148...

%e 3 7 19 49 131 339 899 2344 6180 16040 41572 105806 267560 666682 1655140 4070280...

%e 3 7 19 49 131 339 899 2345 6198 16204 42586 110636 286682 733032 1865008 4693178...

%e 3 7 19 49 131 339 899 2345 6199 16224 42788 112016 293908 764248 1982070 5089002...

%e 3 7 19 49 131 339 899 2345 6199 16225 42810 112260 295734 774682 2030988 5286652...

%e 3 7 19 49 131 339 899 2345 6199 16225 42811 112284 296024 777042 2045610 5360672...

%e 3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296050 777382 2048600 5380646...

%e 3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777410 2048994 5384370...

%e 3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777411 2049024 5384822...

%e 3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777411 2049025 5384854...

%e 3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777411 2049025 5384855...

%e ...

%Y Cf. A116903 (h->infinity), A038577 (h=1), A302408 (h=2), A001411, A038373.

%K nonn,walk,tabl

%O 1,1

%A _Scott R. Shannon_, Aug 04 2020