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, 114482128183138374886637093070429, 2465467527044697154210112460659081

Offset: 1

Author

S. Brunner, Feb 02 2020

Keywords

• nonn
• walk
• hard
• nice

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.

Examples

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

Links

• S. Brunner, Python program.
• Peter Kagey, Example of a(5) = 8.

Crossrefs

Cf. A145157, A265914.

Extensions

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

a(21) from Andrew Howroyd, Oct 16 2024

a(22) from Oliver R. Bellwood, Jul 18 2025