A359709 -id:A359709 - cp's OEIS frontend

A359073 Sum of square end-to-end displacements over all n-step self-avoiding walks of A359709.

0, 4, 16, 44, 160, 556, 1744, 12252, 15840, 98876, 138160, 709900, 1155616, 5098260, 11820656, 37085908, 111147104, 281078764, 932893104, 2255139900, 7295211968, 18928121236, 54864568720, 160016686500, 404167501888, 1331607134172, 2945597090384, 10805511468852, 21448743511648

Offset: 0

Scott R. Shannon, Jan 12 2023

Cf. A359709, A103606, A359133, A336448, A001411, A356617, A173380, A337353, A358036, A358046.

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

Original entry on oeis.org

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

Offset: 0

Scott R. Shannon, Jan 12 2023

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).

			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.

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

Cf. A359133, A001412, A359709, A118313.

cp's OEIS Frontend

A359073 Sum of square end-to-end displacements over all n-step self-avoiding walks of A359709.

Views

Author

Keywords

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs