login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A282801 Binary 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 507", based on the 5-celled von Neumann neighborhood. 4
1, 10, 111, 10, 11111, 1010, 1111111, 101010, 111111111, 10101010, 11111111111, 1010101010, 1111111111111, 101010101010, 111111111111111, 10101010101010, 11111111111111111, 1010101010101010, 1111111111111111111, 101010101010101010, 111111111111111111111 (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, Feb 22 2017: (Start)

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

a(n) = (10^n - 10) / 99 for n>1 and odd.

a(n) = 101*a(n-2) - 100*a(n-4) for n>3.

G.f.: (1 + 10*x + 10*x^2 - 1000*x^3 + 1000*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 = 507; 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. A282800, A282802, A282803.

Sequence in context: A100751 A282452 A282653 * A228006 A004290 A257344

Adjacent sequences:  A282798 A282799 A282800 * A282802 A282803 A282804

KEYWORD

nonn,easy

AUTHOR

Robert Price, Feb 21 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 19 20:42 EST 2019. Contains 329323 sequences. (Running on oeis4.)