# Self-avoiding Walk with next-nearest neighbors

Background

I study polymer physics and am doing experiments testing the model outlined in this paper. Basically, the polymers fall into an integer number of pits, and we create a partition function based on the number as the parameter. The multiplicity of each number-state is based on the number of non-degenerate self-avoiding random walks, from this sequence.

The problem

I've noticed that some of the molecular conformations are not nearest-neighbor self avoiding walks, but include next-nearest-neighbor chains. I want to derive a theory based on next-nearest-neighbor self-avoiding walk, and I need to know the number of unique configurations with N steps. This paper outlines the total number of configurations for m nearest and n next-nearest neighbors, but includes degenerate configurations. I want to know the number of unique configurations for each m and n, not the total, but I can't seem to find it.

The relationship between total and unique for nearest neighbor is $N_{u}=\frac{N+4}{8}$, but that relationship doesn't apply to the next-nearest numbers (many appear to be 2 (mod 6)). Does anyone know how or where I can figure this out, short of drawing every possible configuration on graph paper?

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