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A270937 First differences of number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 221", based on the 5-celled von Neumann neighborhood. 1
7, -4, 40, -35, 107, -107, 211, -211, 347, -347, 515, -515, 715, -715, 947, -947, 1211, -1211, 1507, -1507, 1835, -1835, 2195, -2195, 2587, -2587, 3011, -3011, 3467, -3467, 3955, -3955, 4475, -4475, 5027, -5027, 5611, -5611, 6227, -6227, 6875, -6875, 7555 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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..127

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, Mar 26 2016: (Start)

a(n) = 4-9*(-1)^n+(4+8*(-1)^n)*n+4*(-1)^n*n^2 for n>2.

a(n) = 4*n^2+12*n-5 for n>2 and even.

a(n) = -4*n^2-4*n+13 for n>2 and odd.

a(n) = -a(n-1)+2*a(n-2)+2*a(n-3)-a(n-4)-a(n-5) for n>7.

G.f.: (7+3*x+22*x^2-x^3+7*x^4-7*x^5-4*x^6+5*x^7) / ((1-x)^2*(1+x)^3).

(End)

MATHEMATICA

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

code=221; 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 *)

Table[on[[i+1]]-on[[i]], {i, 1, Length[on]-1}] (* Difference at each stage *)

CROSSREFS

Cf. A270934.

Sequence in context: A270629 A270680 A270904 * A270320 A270332 A038270

Adjacent sequences:  A270934 A270935 A270936 * A270938 A270939 A270940

KEYWORD

sign,easy

AUTHOR

Robert Price, Mar 26 2016

STATUS

approved

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Last modified September 22 09:32 EDT 2020. Contains 337289 sequences. (Running on oeis4.)