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Number of black cells after n moves of Langton's ant on an infinite hexagonal grid, starting with only white cells.
+10
19
0, 1, 2, 3, 4, 5, 6, 5, 6, 7, 8, 9, 8, 7, 8, 9, 10, 11, 10, 9, 10, 11, 12, 13, 12, 13, 14, 15, 16, 17, 18, 17, 16, 17, 18, 19, 20, 19, 18, 19, 20, 21, 22, 21, 20, 19, 18, 19, 20, 21, 22, 21, 20, 21, 22, 23, 24, 23, 22, 21, 20, 21, 22, 23, 24, 23, 22, 23, 24, 25, 26
COMMENTS
On a white cell, turn 60 degrees right, flip the color of the cell, then move forward one unit. On a black cell, turn 60 degrees left, flip the color of the cell, then move forward one unit.
One may see the ant as (1) living on a hexagonal tiling (as in the illustration), in which case one third of all tiles are never visited, or (2) as living on a triangular tiling, in which case these never-visited hexagonal tiles are divided between six neighboring tiles to form triangular tiles, or (3) as living on a hexagonal grid understood as a graph dual to that triangular tiling, in which case the ant travels from one vertex to another using edges. - Andrey Zabolotskiy, Oct 09 2016
Iterations at which Langton's Ant living on triangular tiling passes through the origin.
+10
5
0, 6, 24, 30, 72, 78, 96, 102, 108, 174, 180, 198, 212, 222, 252, 282, 292, 306, 324, 330, 408, 414, 420, 438, 444, 522, 544, 554, 576, 594, 648, 666, 672, 798, 804, 810, 852, 858, 920, 926, 972, 978, 984, 1018, 1024, 1154, 1160, 1178, 1184, 1190, 1208, 1214
COMMENTS
Langton's Ant living on triangular tiling (or, equivalently, hexagonal grid) follows the rules similar to those of the ordinary Langton's ant. On a white cell, turn 60 degrees right, flip the color of the cell, then move forward one unit. On a black cell, turn 60 degrees left, flip the color of the cell, then move forward one unit.
On these iterations pattern becomes symmetric. Orientation of the ant on these iterations is always the same.
Empirically, a(n) ~ c*n^1.207.
Iterations at which Langton's Ant living on triangular tiling reaches the distance of n from the origin for the first time.
+10
4
1, 2, 3, 14, 15, 48, 53, 136, 145, 362, 375, 474, 491, 724, 745, 1904, 1921, 2234, 2267, 2362, 2383, 2500, 2537, 2786, 2811, 3542, 3575, 8304, 8325, 8432, 8501, 8948, 8989, 14858, 14911, 15256, 15309, 18258, 18367, 21804, 22021, 22380, 22453, 23222, 23279
COMMENTS
The distance is defined as the number of steps needed to reach the origin (analog of Manhattan distance).
a(n) ~ c*n^2; however, the first several hundreds of terms are very well described by the approximate formula c'*n^(2.8). [amended by Andrey Zabolotskiy, Oct 09 2016 and Nov 02 2016]
Intervals between iterations at which Langton's Ant living on triangular tiling reaches the distance of n from the origin for the first time.
+10
4
1, 1, 1, 11, 1, 33, 5, 83, 9, 217, 13, 99, 17, 233, 21, 1159, 17, 313, 33, 95, 21, 117, 37, 249, 25, 731, 33, 4729, 21, 107, 69, 447, 41, 5869, 53, 345, 53, 2949, 109, 3437, 217, 359, 73, 769, 57, 10181, 81, 2291, 97, 3217, 73, 6445, 105, 493, 81, 6035, 113
COMMENTS
The distance is defined as the number of steps needed to reach the origin (analog of Manhattan distance). It seems that starting from n=625 (which corresponds to iterations around 26,000,000), a(n)=53 for odd n. [amended by Andrey Zabolotskiy, Oct 09 2016]
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