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.
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
T(1,3) = 22. The five 3-step walks taking a first step to the right and upward or a step upward and then to the right are:
.
+ +--+ +--+ +--+--+ +--+
| | | | | |
X--+--+ X--+ X--+ X X +
.
The same steps can be taken to the right then down, to the left then down, and to the left then up. There is also the two straight walks right and left. This give a total number of walks of 4*5+2 = 22.
.
The table begins:
.
4 10 22 42 90 182 382 742 1486 2866 5646 10878 21198 40694 78758...
4 12 34 82 194 438 1034 2362 5558 12662 29366 66330 151566 339514 767798...
4 12 36 98 262 650 1610 3870 9490 22830 55826 134242 326934 784770 1901246...
4 12 36 100 282 754 1994 5046 12786 31746 79566 196858 491506 1214262 3024890...
4 12 36 100 284 778 2142 5682 14986 38462 98762 249894 635290 1599394 4048366...
4 12 36 100 284 780 2170 5882 15970 42286 111554 288962 748414 1916762 4921146...
4 12 36 100 284 780 2172 5914 16230 43730 117810 311894 823682 2146886 5593690...
4 12 36 100 284 780 2172 5916 16266 44058 119842 321630 862674 2284682 6040622...
4 12 36 100 284 780 2172 5916 16268 44098 120246 324394 877210 2348022 6281498...
4 12 36 100 284 780 2172 5916 16268 44100 120290 324882 880866 2368982 6380418...
4 12 36 100 284 780 2172 5916 16268 44100 120292 324930 881446 2373706 6409762...
4 12 36 100 284 780 2172 5916 16268 44100 120292 324932 881498 2374386 6415746...
4 12 36 100 284 780 2172 5916 16268 44100 120292 324932 881500 2374442 6416534...
4 12 36 100 284 780 2172 5916 16268 44100 120292 324932 881500 2374444 6416594...
4 12 36 100 284 780 2172 5916 16268 44100 120292 324932 881500 2374444 6416596...
Comments