A322567 - OEIS

%I #25 Feb 03 2023 01:32:34

%S 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,

%T 16,17,18,19,18,17,16,15,14,15,14,15,16,17,18,19,20,19,20,21,22,23,24,

%U 25,24,25,26,27,28,27,26,27,28,29,30,31,30,29,30,31,32

%N 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.

%C 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).

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

%C 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).

%H Lars Blomberg, <a href="/A322567/b322567.txt">Table of n, a(n) for n = 0..10000</a>

%H Lars Blomberg, <a href="/A322567/a322567.png">Illustration of the tiling</a>

%H Lars Blomberg, <a href="/A322567/a322567_1.png">State when the highway goes outside the limit</a>

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

%K nonn

%O 0,3

%A _Lars Blomberg_, Aug 29 2019