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A270009 First differences of number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 5", based on the 5-celled von Neumann neighborhood. 2
7, -8, 49, -49, 121, -121, 225, -225, 361, -361, 529, -529, 729, -729, 961, -961, 1225, -1225, 1521, -1521, 1849, -1849, 2209, -2209, 2601, -2601, 3025, -3025, 3481, -3481, 3969, -3969, 4489, -4489, 5041, -5041, 5625, -5625, 6241, -6241, 6889, -6889, 7569 (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 09 2016: (Start)

a(n) = 4+5*(-1)^n+(4+8*(-1)^n)*n+4*(-1)^n*n^2 for n>1.

a(n) = 4*n^2+12*n+9 for n>1 and n even.

a(n) = -4*n^2-4*n-1 for n>1 and n odd.

a(n) = -a(n-1)+2*a(n-2)+2*a(n-3)-a(n-4)-a(n-5) for n>6.

G.f.: (7-x+27*x^2+2*x^3-3*x^4-x^5+x^6) / ((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=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}];

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. A270006.

Sequence in context: A303732 A119453 A047192 * A033044 A025630 A036566

Adjacent sequences:  A270006 A270007 A270008 * A270010 A270011 A270012

KEYWORD

sign,easy

AUTHOR

Robert Price, Mar 08 2016

STATUS

approved

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Last modified August 18 13:25 EDT 2019. Contains 326100 sequences. (Running on oeis4.)