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A277936 Decimal representation of the x-axis, from the left edge to the origin, or from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 7", based on the 5-celled von Neumann neighborhood. 2
1, 3, 0, 15, 0, 63, 0, 255, 0, 1023, 0, 4095, 0, 16383, 0, 65535, 0, 262143, 0, 1048575, 0, 4194303, 0, 16777215, 0, 67108863, 0, 268435455, 0, 1073741823, 0, 4294967295, 0, 17179869183, 0, 68719476735, 0, 274877906943, 0, 1099511627775, 0, 4398046511103, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Initialized with a single black (ON) cell at stage zero.

Essentially the same as A277929 and A277928. - R. J. Mathar, Nov 09 2016

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 the 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

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 06 2016: (Start)

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

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

a(n) = (-1/2-(-2)^n+(-1)^n/2+2^n) for n>0. (End)

MAPLE

A277936:=n->(-1/2-(-2)^n+(-1)^n/2+2^n): 1, seq(A277936(n), n=1..80); # Wesley Ivan Hurt, Jan 24 2017

MATHEMATICA

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

code=7; 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[1, i]], 2], {i, 1, stages-1}]

CROSSREFS

Cf. A277560.

Sequence in context: A275831 A065121 A167339 * A138540 A123023 A130637

Adjacent sequences:  A277933 A277934 A277935 * A277937 A277938 A277939

KEYWORD

nonn,easy

AUTHOR

Robert Price, Nov 05 2016

STATUS

approved

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Last modified March 20 07:27 EDT 2019. Contains 321345 sequences. (Running on oeis4.)