A359709 Number of n-step self-avoiding walks on a 2D square lattice whose end-to-end distance is an integer.

1, 4, 4, 12, 28, 76, 164, 732, 1044, 4924, 6724, 30636, 43972, 190516, 313996, 1197908, 2284260, 7678188, 16257604, 50524252, 113052396, 341811828, 773714436, 2358452388, 5245994292, 16447462492, 35395532236, 115129727188, 238542983748, 804980005276

Offset

0,2

Comments

The walks counted are all those directly along and x or y axes, and all walks whose final (|x|,|y|) lattice point are the two legs of a Pythagorean triple.

Links

Table of n, a(n) for n=0..29.

Wikipedia, Self-avoiding walk.

Example

a(3) = 12 as, in the first quadrant, there is one 3-step SAW whose end-to-end distance is an integer (1 unit):
.
     X---.
         |
     X---.
.
This can be walked in 8 different ways on a 2D square lattice. There are also the four walks directly along the x and y axes, giving a total of 8 + 4 = 12 walks.

Crossrefs

Cf. A359073, A103606, A359741, A001411, A356617, A173380, A337353, A358036, A358046.

Sequence in context: A303644 A298796 A106232 * A228612 A038804 A183362

Adjacent sequences: A359706 A359707 A359708 * A359710 A359711 A359712

Keyword

nonn,walk

Author

Scott R. Shannon, Jan 12 2023

Status

approved