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A282124 Decimal 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 430", based on the 5-celled von Neumann neighborhood. 4
1, 3, 3, 15, 11, 63, 43, 255, 171, 1023, 683, 4095, 2731, 16383, 10923, 65535, 43691, 262143, 174763, 1048575, 699051, 4194303, 2796203, 16777215, 11184811, 67108863, 44739243, 268435455, 178956971, 1073741823, 715827883, 4294967295, 2863311531, 17179869183 (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 Colin Barker, Feb 07 2017: (Start)
a(n) = (-1 + 2*(-1)^n - (-1)^n*2^(1+n) + 2^(2+n)) / 3.
a(n) = 5*a(n-2) - 4*a(n-4) for n>3.
G.f.: (1 + 3*x - 2*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
(End)
Conjectures from Paul Curtz, Jun 10 2019: (Start)
a(n) = A001045(n+1)*(period 2: repeat[1, 3]).
a(n+4) = a(n) + 10*A081631(n).
a(2*n+1) = 2^(2*n+2) -1.
a(n+2) = a(n) + A098646(n+1).
(End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 430; 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]], 2], {i , 1, stages - 1}]
CROSSREFS
Sequence in context: A160612 A275324 A282663 * A281421 A280840 A212200
KEYWORD
nonn,easy
AUTHOR
Robert Price, Feb 06 2017
STATUS
approved

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Last modified March 29 10:59 EDT 2024. Contains 371277 sequences. (Running on oeis4.)