A331001 - OEIS

%I #44 Feb 21 2020 11:10:24

%S 1,1,1,2,8,24,282,888,46933,238119,36027060,187011538,130162111969,

%T 1084873972934,2200211600730504,18559765767843341,

%U 174907641314142138422,2355130982684196593401,65250573687646264926302133,884112393542714503429381555

%N Number of symmetrical self-avoiding walks with maximum length on an n X n board which start in the upper left corner and go right on the first step.

%C If you allow going down on the first step you get two times a(n) for n > 1.

%C All symmetrical self-avoiding walks on a square board with odd length seem to have a 180-degree rotational symmetry, and all symmetrical self-avoiding walks on a square board with even length seem to have either vertically or horizontally reflection symmetry.

%H S. Brunner, <a href="https://pastebin.com/9kxPM2hF">Python program</a>.

%H Peter Kagey, <a href="/A331001/a331001.pdf">Example of a(5) = 8</a>.

%e The solutions for n=3 and n=4:

%e n=3:  |    n=4:

%e 1     |    1          2

%e >>v   |   >>>v   |   >v>

%e v<<   |   v<<<   |   v<^<

%e >>    |   >>>v   |   v>v^

%e       |    <<<   |   >^>^

%Y Cf. A145157, A265914.

%K nonn,walk,hard,nice

%O 1,4

%A _S. Brunner_, Feb 02 2020

%E a(11)-a(20) from _Andrew Howroyd_, Feb 20 2020