login
A359741
Number of n-step self-avoiding walks on a 3D cubic lattice whose end-to-end distance is an integer.
3
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).
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
KEYWORD
nonn,walk,more
AUTHOR
Scott R. Shannon, Jan 12 2023
STATUS
approved