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A283605 Binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 670", based on the 5-celled von Neumann neighborhood. 4
1, 11, 111, 1111, 10111, 111111, 1110111, 11111111, 101110111, 1111111111, 11011110111, 111111111111, 1100111110111, 11111111111111, 110111111110111, 1111111111111111, 11111111111110111, 111111111111111111, 1111111111111110111, 11111111111111111111 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Initialized with a single black (ON) cell at stage zero.
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
LINKS
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
FORMULA
Conjectures from Chai Wah Wu, May 06 2024: (Start)
a(n) = 10*a(n-1) + a(n-2) - 10*a(n-3) for n > 17.
G.f.: (-10000000000000*x^17 + 1000000000000*x^16 + 9890000000000*x^15 - 989000000000*x^14 + 109000000000*x^13 - 10900000000*x^12 + 900000000*x^11 - 90000000*x^10 + 100000000*x^9 - 10000000*x^8 + 10000*x^5 - 1000*x^4 + x + 1)/((x - 1)*(x + 1)*(10*x - 1)). (End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 670; 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[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
CROSSREFS
Sequence in context: A284088 A284420 A283850 * A083440 A138144 A284243
KEYWORD
nonn,easy
AUTHOR
Robert Price, Mar 11 2017
STATUS
approved

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Last modified June 15 13:08 EDT 2024. Contains 373407 sequences. (Running on oeis4.)