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

%I #16 Jan 15 2023 15:11:26

%S 1,6,6,30,78,1134,1350,20574,23238,390606,496998,7614750,10987926,

%T 152120934,237122526,3110708214,5017927638,64718847438,105210653478,

%U 1362453235998

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

%C 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).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Self-avoiding_walk">Self-avoiding walk</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Pythagorean_quadruple">Pythagorean quadruple</a>.

%e a(3) = 30 as, in the first octant, there is one 3-step SAW whose end-to-end distance is an integer (1 unit):

%e .

%e X---.

%e |

%e X---.

%e .

%e 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.

%Y Cf. A359133, A001412, A359709, A118313.

%K nonn,walk,more

%O 0,2

%A _Scott R. Shannon_, Jan 12 2023