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A277926 Binary representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 5", based on the 5-celled von Neumann neighborhood. 4
1, 10, 0, 1111, 0, 111111, 0, 11111111, 0, 1111111111, 0, 111111111111, 0, 11111111111111, 0, 1111111111111111, 0, 111111111111111111, 0, 11111111111111111111, 0, 1111111111111111111111, 0, 111111111111111111111111, 0, 11111111111111111111111111, 0 (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

Index entries for sequences related to cellular automata

Index to 2D 5-Neighbor Cellular Automata

Index to Elementary Cellular Automata

Robert Price, Diagrams of first 20 stages

FORMULA

Conjectures from Colin Barker, Nov 04 2016: (Start)

a(n) = 0 for n>1 and even; a(n) = (10^(n+1)-1)/9 for n>1 and odd.

a(n) = 101*a(n-2)-100*a(n-4) for n>5.

G.f.: (1+10*x-101*x^2+101*x^3+100*x^4-100*x^5) / ((1-x)*(1+x)*(1-10*x)*(1+10*x)).

(End)

MATHEMATICA

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

code=5; 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]], 10], {i, 1, stages-1}]

CROSSREFS

Cf. A277927, A277928, A277929.

Sequence in context: A286403 A287129 A287190 * A222521 A068159 A108695

Adjacent sequences:  A277923 A277924 A277925 * A277927 A277928 A277929

KEYWORD

nonn,easy

AUTHOR

Robert Price, Nov 04 2016

STATUS

approved

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Last modified December 6 01:46 EST 2021. Contains 349558 sequences. (Running on oeis4.)