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.
%I A323131 #15 Jan 18 2019 14:41:47 %S A323131 1,7,47,303,1921,11963,74130,454484,2779152,16882278,102384151, %T A323131 618136584,3727827148,22408576099,134595908277,806452390868 %N A323131 Number of uncrossed rooted knight's paths of length n on an infinite board. %C A323131 The direction of the first move is kept fixed. %C A323131 The average number of steps of a random walk using such knight moves with forbidden crossing is 45 (compare to A322831). %H A323131 Hugo Pfoertner, <a href="http://www.randomwalk.de/sequences/a323131.htm">Illustrations of rooted uncrossed knight's paths of length <= 3</a>, (2019). %e A323131 a(1) = 1: The fixed initial move. %e A323131 a(2) = 7: Relative to the direction given by the initial move, there are 7 possible direction changes. The backward direction is illegal for the self-avoiding uncrossed path. Only for the right angle turn its mirror image would coincide with the turn in the opposite direction. Therefore this move would be eliminated in the unrooted walks, making a(2) > A323132(2) = 6. %e A323131 a(3) = 47: 2 of all 7*7 = 49 continuation moves already lead to a crossing of the first path segment. %e A323131 See illustrations at Pfoertner link. %Y A323131 Cf. A003192, A272773, A323132, A323133, A323134, A323559. %K A323131 nonn,walk,more,hard %O A323131 1,2 %A A323131 _Hugo Pfoertner_, Jan 05 2019 %E A323131 Erroneous (as pointed out by _Bert Dobbelaere_) a(8) and a(10) corrected by _Hugo Pfoertner_, Jan 18 2019 %E A323131 a(12)-a(16) from _Bert Dobbelaere_, Jan 18 2019