Revision History for A280373
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| A280373
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Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 260", based on the 5-celled von Neumann neighborhood.
(history;
published version)
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#10 by Bruno Berselli at Mon Jan 02 05:56:02 EST 2017
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#9 by Colin Barker at Mon Jan 02 05:40:57 EST 2017
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#8 by Colin Barker at Mon Jan 02 05:40:45 EST 2017
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| FORMULA
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Conjectures from Colin Barker, Jan 02 2017: (Start)
a(n) = 16*a(n-4) for n>3.
G.f.: (1 + 2*x + 5*x^2 + 8*x^3 + 8*x^5 + 4*x^6 + 2*x^7 + x^8 - 16*x^12) / ((1 - 2*x)*(1 + 2*x)*(1 + 4*x^2)).
(End)
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approved
editing
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#7 by Bruno Berselli at Sun Jan 01 14:36:33 EST 2017
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#6 by Robert Price at Sun Jan 01 11:38:18 EST 2017
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#5 by Robert Price at Sun Jan 01 11:38:16 EST 2017
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#4 by Robert Price at Sun Jan 01 11:35:42 EST 2017
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| LINKS
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Robert Price, <a href="/A280373/a280373.tmp.txt">Diagrams of first 20 stages</a>
Robert Price, <a href="/A280373/a280373.tmp.txt">Diagrams of first 20 stages</a>
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#3 by Robert Price at Sun Jan 01 11:35:29 EST 2017
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| LINKS
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Robert Price, <a href="/A280373/b280373.txt">Table of n, a(n) for n = 0..126</a>
Robert Price, <a href="/A280373/a280373.tmp.txt">Diagrams of first 20 stages</a>
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#2 by Robert Price at Sun Jan 01 11:33:46 EST 2017
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| NAME
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allocatedDecimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 260", based on the 5-celled forvon RobertNeumann Priceneighborhood.
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| DATA
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1, 2, 5, 8, 16, 40, 84, 130, 257, 640, 1344, 2080, 4096, 10240, 21504, 33280, 65536, 163840, 344064, 532480, 1048576, 2621440, 5505024, 8519680, 16777216, 41943040, 88080384, 136314880, 268435456, 671088640, 1409286144, 2181038080, 4294967296, 10737418240
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| OFFSET
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0,2
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| COMMENTS
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Initialized with a single black (ON) cell at stage zero.
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| REFERENCES
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S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
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| LINKS
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N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
<a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
<a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
<a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
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| MATHEMATICA
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CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 260; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0, Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]], Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[1, i]], 2], {i, 1, stages - 1}]
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| KEYWORD
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allocated
nonn,easy
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| AUTHOR
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Robert Price, Jan 01 2017
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| STATUS
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approved
editing
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#1 by Robert Price at Sun Jan 01 11:33:46 EST 2017
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| NAME
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allocated for Robert Price
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| KEYWORD
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allocated
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| STATUS
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approved
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