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A287131 Decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 245", based on the 5-celled von Neumann neighborhood. 4
1, 2, 0, 14, 3, 62, 3, 252, 3, 1020, 3, 4092, 3, 16380, 3, 65532, 3, 262140, 3, 1048572, 3, 4194300, 3, 16777212, 3, 67108860, 3, 268435452, 3, 1073741820, 3, 4294967292, 3, 17179869180, 3, 68719476732, 3, 274877906940, 3, 1099511627772, 3, 4398046511100, 3 (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

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, May 20 2017: (Start)

G.f.: (1 + 2*x - 5*x^2 + 4*x^3 + 7*x^4 - 12*x^6 - 2*x^7 + 8*x^9) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).

a(n) = 3 for n>5 and even.

a(n) = 2^(n+1) - 4 for n>5 and odd.

(End)

MATHEMATICA

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

code = 245; 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. A287129, A287130, A287132.

Sequence in context: A286772 A286505 A286405 * A287192 A219843 A064855

Adjacent sequences:  A287128 A287129 A287130 * A287132 A287133 A287134

KEYWORD

nonn,easy

AUTHOR

Robert Price, May 20 2017

STATUS

approved

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Last modified March 2 19:18 EST 2021. Contains 341756 sequences. (Running on oeis4.)