A255938 Langton's ant walk: number of black cells on the infinite grid after the ant moves n times.

0, 1, 2, 3, 4, 3, 4, 5, 6, 7, 6, 7, 8, 9, 10, 9, 8, 7, 6, 7, 6, 7, 8, 9, 10, 9, 10, 11, 12, 13, 12, 11, 10, 9, 10, 9, 10, 11, 12, 13, 12, 13, 14, 15, 16, 15, 14, 13, 12, 13, 12, 11, 12, 13, 12, 13, 14, 15, 16, 15, 14, 13, 12, 13, 12, 13, 14, 15, 16, 15, 16, 17

Offset: 0

Author

Arkadiusz Wesolowski, Mar 11 2015

Keywords

• nonn
• easy

Comments

The ant starts from a completely white grid.

From Albert Lau, Jun 19 2016: (Start)

After n steps, the direction in which the ant is facing is 90 degree * a(n). For each 360 degrees, the ant makes a full turn.

The ant's position after n steps is Sum_{k=1..n} e^(a(n)*i*Pi/2) when expressed as a complex number. (End)

References

• D. Gale, Tracking the Automatic Ant and Other Mathematical Explorations, A Collection of Mathematical Entertainments Columns from The Mathematical Intelligencer, Springer, 1998; see p. 63.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..20000
• A. Gajardo, A. Moreira, and E. Goles, Complexity of Langton's ant, Discrete Applied Mathematics, 117 (2002), 41-50.
• Chris G. Langton, Studying artificial life with cellular automata, Physica D: Nonlinear Phenomena, 22 (1-3) (1986), 120-149.
• Wikipedia, Langton's ant.

Crossrefs

Cf. A126978.

Programs

Mathematica

size = 10;
grid = SparseArray[{}, {size, size}, 1];
{X, Y, n} = {size, size, 0}/2 // Round;
While[1 <= X <= size && 1 <= Y <= size,
     n += grid[[X, Y]] // Sow;
     grid[[X, Y]] *= -1;
     {X, Y} += {Cos[\[Pi]/2 n], Sin[\[Pi]/2 n]};
     ] // Reap // Last // Last // Prepend[#, 0] &
(* Albert Lau, Jun 19 2016 *)

Formula

a(n+104) = a(n) + 12 for n > 9976. - Andrey Zabolotskiy, Jul 05 2016