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A270010 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 7", based on the 5-celled von Neumann neighborhood. 3
1, 9, 0, 49, 0, 121, 0, 225, 0, 361, 0, 529, 0, 729, 0, 961, 0, 1225, 0, 1521, 0, 1849, 0, 2209, 0, 2601, 0, 3025, 0, 3481, 0, 3969, 0, 4489, 0, 5041, 0, 5625, 0, 6241, 0, 6889, 0, 7569, 0, 8281, 0, 9025, 0, 9801, 0, 10609, 0, 11449, 0, 12321, 0, 13225, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Initialized with a single black (ON) cell at stage zero.

Seems to differ from A270006 only at a(1). - R. J. Mathar, Mar 21 2016

REFERENCES

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

LINKS

Robert Price, Table of n, a(n) for n = 0..128

Robert Price, Diagrams of the 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

FORMULA

Conjectures from Colin Barker, Mar 09 2016: (Start)

a(n) = (-1)*((-1+(-1)^n)*(1+2*n)^2)/2 for n>0.

a(n) = 0 for n>0 and even.

a(n) = 4*n^2+4*n+1 for n odd.

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

G.f.: (1+9*x-3*x^2+22*x^3+3*x^4+x^5-x^6) / ((1-x)^3*(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=7; 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}];

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

CROSSREFS

Sequence in context: A182213 A340954 A087094 * A167319 A231948 A222396

Adjacent sequences:  A270007 A270008 A270009 * A270011 A270012 A270013

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 08 2016

STATUS

approved

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Last modified March 8 01:41 EST 2021. Contains 341934 sequences. (Running on oeis4.)