This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
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:
.
+ +--+ +--+ +--+--+ +--+
| | | | | |
+--+--+--+ +--+--+ +--+ +--+ + + +
.
The same steps can be taken to the left, giving a total of 2*6 = 12 walks.
.
The table begins:
.
3 6 12 20 36 58 100 160 268 430 708 1140 1860 3002 4876 7880...
3 7 18 40 86 170 350 688 1394 2702 5338 10278 20078 38578 74820 143496...
3 7 19 48 120 274 620 1346 2972 6402 13994 29870 64412 136308 291008 612920...
3 7 19 49 130 326 810 1912 4486 10262 23634 53642 122624 276524 627248 1405154...
3 7 19 49 131 338 884 2228 5560 13438 32320 76440 181202 425138 1001128 2336886...
3 7 19 49 131 339 898 2328 6050 15320 38478 94642 231798 560794 1357098 3258148...
3 7 19 49 131 339 899 2344 6180 16040 41572 105806 267560 666682 1655140 4070280...
3 7 19 49 131 339 899 2345 6198 16204 42586 110636 286682 733032 1865008 4693178...
3 7 19 49 131 339 899 2345 6199 16224 42788 112016 293908 764248 1982070 5089002...
3 7 19 49 131 339 899 2345 6199 16225 42810 112260 295734 774682 2030988 5286652...
3 7 19 49 131 339 899 2345 6199 16225 42811 112284 296024 777042 2045610 5360672...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296050 777382 2048600 5380646...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777410 2048994 5384370...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777411 2049024 5384822...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777411 2049025 5384854...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777411 2049025 5384855...
...
a(0) = 1, as there is one self-avoiding walk of length 0, namely the null-walk (the walk whose steps are the null set).
a(1) = 4 because (using the terminology in the Comment), the 4 possible 1-step walks are W_1 = {l,r,c,c-}.
a(2) = 12 because the set of legal 2-step walks are {l^2, lc, lc-, r^2, rc, rc-, c^2, cl, cr, c^-2, c-l, c-r}.
a(3) = 34 because we have every W_2 concatenated with {l,r,c,c-} except for those with immediate violations (lr etc.) and those two which go in a triangle {c^3, c^-3}; hence a(3) = 3*a(2) - 2 = 3*12 - 2 = 36 - 2 = 34.
w = [[[(0, 0)]]]
for n in range(1, 15):
nw = []
for walk in w[-1]:
(x, t) = walk[-1]
nss = [(x-1, t), (x+1, t), (x, (t+1)%3), (x, (t-1)%3)]
for ns in nss:
if ns not in walk:
nw.append(walk[:] + [ns])
w.append(nw)
print([len(x) for x in w])
# Andrey Zabolotskiy, Sep 19 2019
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