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A270985 Number of active (ON,black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 238", based on the 5-celled von Neumann neighborhood. 0
1, 5, 20, 76, 300, 1212, 4900, 19740, 79284, 317836, 1272804, 5094204, 20382868, 81543660, 326199108, 1304845468 (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

Table of n, a(n) for n=0..15.

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, Jun 07 2016: (Start)

a(n) = (320-27*2^(3+n)+175*4^n+384*n)/144 for n>2.

a(n) = 8*a(n-1)-21*a(n-2)+22*a(n-3)-8*a(n-4) for n>6.

G.f.: (1+x)*(1-4*x+5*x^2-6*x^3+16*x^4-8*x^5) / ((1-x)^2*(1-2*x)*(1-4*x)).

(End)

MATHEMATICA

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

code=238; 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}];

on=Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)

Part[on, 2^Range[0, Log[2, stages]]] (* Extract relevant terms *)

CROSSREFS

Cf. A270984.

Sequence in context: A300918 A269708 A295347 * A289786 A129869 A271887

Adjacent sequences:  A270982 A270983 A270984 * A270986 A270987 A270988

KEYWORD

nonn,more

AUTHOR

Robert Price, Mar 27 2016

EXTENSIONS

a(8)-a(15) from Lars Blomberg, Jun 07 2016

STATUS

approved

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Last modified November 12 04:21 EST 2019. Contains 329051 sequences. (Running on oeis4.)