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A269908 Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 1", based on the 5-celled von Neumann neighborhood. 1
1, 5, 6, 50, 51, 167, 168, 388, 389, 745, 746, 1270, 1271, 1995, 1996, 2952, 2953, 4173, 4174, 5690, 5691, 7535, 7536, 9740, 9741, 12337, 12338, 15358, 15359, 18835, 18836, 22800, 22801, 27285, 27286, 32322, 32323, 37943, 37944, 44180, 44181, 51065, 51066 (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..128

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 08 2016: (Start)

a(n) = (3+9*(-1)^n-2*(1+12*(-1)^n)*n-12*(-2+(-1)^n)*n^2+8*n^3)/12.

a(n) = (4*n^3+6*n^2-13*n+6)/6 for n even.

a(n) = (4*n^3+18*n^2+11*n-3)/6 for n odd.

a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7) for n>6.

G.f.: (1+4*x-2*x^2+32*x^3+x^4-4*x^5) / ((1-x)^4*(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=1; 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[Total[Part[on, Range[1, i]]], {i, 1, Length[on]}] (* Sum at each stage *)

CROSSREFS

Cf. A269906.

Sequence in context: A273050 A163481 A298376 * A157805 A256291 A299243

Adjacent sequences:  A269905 A269906 A269907 * A269909 A269910 A269911

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 07 2016

STATUS

approved

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Last modified July 20 01:17 EDT 2019. Contains 325168 sequences. (Running on oeis4.)