A309384 Number of black squares after n moves of a variant of Langton's ant with turns of 45 degrees.

0, 1, 2, 3, 4, 5, 6, 7, 8, 7, 8, 9, 10, 11, 12, 11, 10, 11, 12, 13, 14, 15, 14, 13, 14, 15, 16, 17, 18, 19, 20, 21, 20, 21, 22, 23, 24, 25, 24, 23, 24, 25, 26, 25, 24, 25, 26, 27, 28, 29, 28, 29, 30, 29, 30, 31, 30, 31, 32, 33, 34, 35, 34, 35, 36, 37, 36, 37

Offset

0,3

Comments

Initially, all squares are white, and the ant is at the origin and oriented towards the east. The ant can travel horizontally or vertically (in any of the four cardinal directions) and diagonally (in any of the four intercardinal directions).

At a white square, the ant turns 45 degrees left, flips the color of the square and moves forward.

At a black square, the ant turns 45 degrees right, flips the color of the square and moves forward.

As in the original variant, the ant eventually builds a recurrent highway pattern, in the present case of 196 steps.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Rémy Sigrist, Representation of the squares visited during the first 100000000 moves (the dark squares are the most visited)

Rémy Sigrist, Representation of the grid after 1021259426 steps

Rémy Sigrist, PARI program for A309384

Wikipedia, Langton's ant

Formula

a(n + 196) = a(n) + 72 for n >= 1021254426.

Example

The first positions of the ant are:
  .  .  4  3  .  .  .  .
  .  5  .  .  2  .  .  .
  .  6  .  .  1  .  .  .
  .  .  7 0,8 .  . 11  .
  .  .  .  .  9 10  .  .

Prog

(PARI) See Links section.

Crossrefs

Cf. A255938.

Sequence in context: A017893 A017883 A269364 * A245353 A063278 A355459

Adjacent sequences: A309381 A309382 A309383 * A309385 A309386 A309387

Keyword

nonn

Author

Rémy Sigrist, Jul 27 2019

Status

approved