A322567 Langton's ant on a tiling with vertex types (3.12.12; 3.4.3.12): number of black cells after n moves of the ant when starting on a dodecagon and looking towards an edge where it meets another dodecagon.

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

Offset

0,3

Comments

Rules for Langton's ant on edge-to-edge tilings by regular polygons: Initially, all tiles are white. On a white tile turn right, on a black tile turn left. Always flip the color of the tile, then move forward one unit. The turn angle for (triangle, square, hexagon, octagon, dodecagon) is (60, 90, 60, 45, 30).

At n=66285 the ant starts a highway along the SE diagonal.

Many other starting positions are possible which give different terms. While they all eventually enter the highway a(n+39) = a(n)+13, the generation where it happens varies (3328, 25256, 66285, 177723, 255119, 354465, 415327).

Links

Lars Blomberg, Table of n, a(n) for n = 0..10000

Lars Blomberg, Illustration of the tiling

Lars Blomberg, State when the highway goes outside the limit

Formula

a(n+39) = a(n)+13 for n > 66285.

Crossrefs

Sequence in context: A268233 A309241 A065651 * A349049 A221356 A177329

Adjacent sequences: A322564 A322565 A322566 * A322568 A322569 A322570

Keyword

nonn

Author

Lars Blomberg, Aug 29 2019

Status

approved