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

Author

Scott R. Shannon, Jan 12 2023

Keywords

• nonn
• walk
• more

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).

Examples

			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.
		

Links

• Wikipedia, Self-avoiding walk.
• Wikipedia, Pythagorean quadruple.

Crossrefs

Cf. A359133, A001412, A359709, A118313.