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A283217 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 597", based on the 5-celled von Neumann neighborhood. 4
1, 1, 1, 13, 17, 29, 17, 221, 273, 477, 273, 3549, 4369, 7645, 4369, 56797, 69905, 122333, 69905, 908765, 1118481, 1957341, 1118481, 14540253, 17895697, 31317469, 17895697, 232644061, 286331153, 501079517, 286331153, 3722304989, 4581298449, 8017272285 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
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, Mar 04 2017: (Start)
G.f.: (1 + x + 12*x^3) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)*(1 + 4*x^2)).
a(n) = a(n-2) + 16*a(n-4) - 16*a(n-6) for n>5.
(End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 597; 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: A283135 A052491 A283294 * A283390 A078138 A164074
KEYWORD
nonn,easy
AUTHOR
Robert Price, Mar 03 2017
STATUS
approved

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Last modified March 28 10:55 EDT 2024. Contains 371241 sequences. (Running on oeis4.)