Gaussian integers near the curve z = exp(I * t).
0, 1, 1, 0, 1, 1, 2, 1, 2, 2, 1, 2, 1, 3, 1, 4, 0, 4, 0, 5, -1, 5, -1, 6, -2, 6, -3, 6, -3, 7, -4, 7, -5, 7, -6, 7, -7, 7, -8, 7, -9, 7, -10, 7, -11, 7, -12, 7, -13, 7, -13, 6, -14, 6, -15, 6, -16, 6, -16, 5, -17, 5, -17, 4, -18, 4, -19, 4, -19, 3, -20, 3, -20, 2
1
All points (x,y) with x and y integer which are within 1/2 unit of the curve are included. We calculate and display only the first rotation (2*Pi) of the curve.
T. D. Noe, Plot of 972 pairs
T. D. Noe, Table of 972 pairs
Eric W. Weisstein, MathWorld: Gaussian Integer
(Mma) t = {{0, 1}}; Do[z = Floor[{Cos[x] Exp[x] + 1/2, Sin[x] Exp[x] + 1/2}]; If[z != t[[-1]], AppendTo[t, z]], {x, 0, 2 Pi, 0.000001}]
Cf. S000390.
sign
T. D. Noe, Dec 02 2014