# Counting the number of back-tracking closed walks on a 2D grid.

Given a reference node at an infinitely spanning 2D grid, how can I express the number of back-tracking closed walks (closed walks that does not contain cycles) for that node in terms of L, which is the length of the walk?

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I would calculate it for $L=2,4,6$ and then look it up in the Online Encyclopedia of Integer Sequences. –  Gerry Myerson Jul 5 '12 at 1:22

I think this is the same as the number of walks of length $2n$ on the 4-regular tree beginning and ending at some fixed vertex, which is tabulated at the Online Encyclopedia of Integer Sequences. The generating function is given as $${3\over1+2\sqrt{1-12x}}$$ and there's also an integral representation, $${2\over\pi}\int_0^{12}{x^n\sqrt{x(12-x)}\over16-x}\,dx$$ and some other formulas.

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