Geodesic
Partition Functions and Spiralling in Plane Random Walk
Canadian mathematical bulletin, Tome 6 (1963) no. 2, pp. 231-237

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Consider the plane symmetric random walk on a square lattice: a particle is initially at the origin in the xy-plane, it makes n consecutive steps of unit length, and each step is made with the probability 1/4 in each one of the four directions parallel to the axes. We call the path of the particle a spiral if the following conditions are met: a) the particle never occupies the same position twice, b) the path of the particle, whenever it turns, either turns always clockwise or always counter-clockwise throughout the path, and c) for every m > n the given n-step path can be continued in atleast one way to give an m-step path meeting the conditions. 
Melzak, Z.A. Partition Functions and Spiralling in Plane Random Walk. Canadian mathematical bulletin, Tome 6 (1963) no. 2, pp. 231-237. doi: 10.4153/CMB-1963-021-3
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     title = {Partition {Functions} and {Spiralling} in {Plane} {Random} {Walk}},
     journal = {Canadian mathematical bulletin},
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     year = {1963},
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