A331001 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.

1, 1, 1, 2, 8, 24, 282, 888, 46933, 238119, 36027060, 187011538, 130162111969, 1084873972934, 2200211600730504, 18559765767843341, 174907641314142138422, 2355130982684196593401, 65250573687646264926302133, 884112393542714503429381555

Offset

1,4

Comments

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

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.

Links

Table of n, a(n) for n=1..20.

S. Brunner, Python program.

Peter Kagey, Example of a(5) = 8.

Example

The solutions for n=3 and n=4:
n=3:  |    n=4:
1     |    1          2
>>v   |   >>>v   |   >v>
v<<   |   v<<<   |   v<^<
>>    |   >>>v   |   v>v^
      |    <<<   |   >^>^

Crossrefs

Cf. A145157, A265914.

Sequence in context: A071599 A047695 A093842 * A050971 A118855 A009515

Adjacent sequences: A330998 A330999 A331000 * A331002 A331003 A331004

Keyword

nonn,walk,hard,nice

Author

S. Brunner, Feb 02 2020

Extensions

a(11)-a(20) from Andrew Howroyd, Feb 20 2020

Status

approved