`` Consider a walk in the square lattice where the distribution of the fundamental steps is [[[1, 0], 1/8], [[-1, 0], 1/8], [[0, 1], 1/8], [[0, -1], 1/8], [[1, 1], 1/8], [[-1, 1], 1/8], [[1, -1], 1/8], [[-1, -1], 1/8]] Let a(n) be the probability of being at the origin after n steps. the probability of returning to the origin after n steps . The linear recurrence operator annihilating a_n is: 2 3 2 (3 n + 7) (n + 1) (27 n + 144 n + 248 n + 139) N - --------------------- - -------------------------------- 2 2 16 (n + 3) (3 n + 4) 16 (n + 3) (3 n + 4) 2 (3 n + 5) (n + 2) (3 n + 8) N 3 - ------------------------------ + N 2 8 (n + 3) (3 n + 4) The whole thing took, 117.887, seconds of CPU time