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A287193 Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 253", based on the 5-celled von Neumann neighborhood. 4
1, 1, 0, 7, 24, 31, 96, 127, 384, 511, 1536, 2047, 6144, 8191, 24576, 32767, 98304, 131071, 393216, 524287, 1572864, 2097151, 6291456, 8388607, 25165824, 33554431, 100663296, 134217727, 402653184, 536870911, 1610612736, 2147483647, 6442450944, 8589934591 (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

Robert Price, Table of n, a(n) for n = 0..126

Robert Price, Diagrams of first 20 stages

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

S. Wolfram, A New Kind of Science

Wolfram Research, Wolfram Atlas of Simple Programs

Index entries for sequences related to cellular automata

Index to 2D 5-Neighbor Cellular Automata

Index to Elementary Cellular Automata

FORMULA

Conjectures from Colin Barker, Nov 22 2017: (Start)

G.f.: (1 + x - 5*x^2 + 2*x^3 + 28*x^4 - 24*x^6) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).

a(n) = (1/4)*(-2 + (-2)^n + 2*(-1)^n + 5*2^n) for n>2.

a(n) = 5*a(n-2) - 4*a(n-4) for n>4.

(End)

MATHEMATICA

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];

code = 253; 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

Cf. A287190, A287191, A287192.

Sequence in context: A077035 A076602 A287132 * A076673 A270422 A063165

Adjacent sequences:  A287190 A287191 A287192 * A287194 A287195 A287196

KEYWORD

nonn,easy

AUTHOR

Robert Price, May 21 2017

STATUS

approved

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Last modified January 17 03:16 EST 2022. Contains 350377 sequences. (Running on oeis4.)