A359741 Number of n-step self-avoiding walks on a 3D cubic lattice whose end-to-end distance is an integer.

1, 6, 6, 30, 78, 1134, 1350, 20574, 23238, 390606, 496998, 7614750, 10987926, 152120934, 237122526, 3110708214, 5017927638, 64718847438, 105210653478, 1362453235998

Offset

0,2

Comments

The walks counted are all those directly along and x, y or z axes, and all walks whose final (x,y,z) lattice point is a solution to the Pythagorean quadruple x^2 + y^2 + z^2 = t^2. The first such solution with all coordinates > 0 is 1^2 + 2^2 + 2^2 = 3^2, which explains the large increase in the number of walks from a(4) to a(5).

Links

Table of n, a(n) for n=0..19.

Wikipedia, Self-avoiding walk.

Wikipedia, Pythagorean quadruple.

Example

a(3) = 30 as, in the first octant, there is one 3-step SAW whose end-to-end distance is an integer (1 unit):
.
     X---.
         |
     X---.
.
This can be walked in 24 different ways on a 3D cubic lattice. There are also the six walks directly along the x, y and z axes, giving a total of 24 + 6 = 30 walks.

Crossrefs

Cf. A359133, A001412, A359709, A118313.

Sequence in context: A377186 A321528 A074002 * A140959 A015699 A244416

Adjacent sequences: A359738 A359739 A359740 * A359742 A359743 A359744

Keyword

nonn,walk,more

Author

Scott R. Shannon, Jan 12 2023

Status

approved