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A270100 Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 81", based on the 5-celled von Neumann neighborhood. 1
1, 5, 10, 50, 59, 159, 180, 376, 405, 725, 766, 1242, 1295, 1935, 2024, 2868, 2985, 4081, 4210, 5590, 5731, 7403, 7580, 9584, 9781, 12149, 12366, 15130, 15375, 18503, 18824, 22356, 22761, 26817, 27186, 31838, 32179, 37395, 37744, 43568, 43929, 50341, 50770 (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

MATHEMATICA

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

code=81; 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. A270098.

Sequence in context: A270089 A336582 A271275 * A271287 A003587 A268100

Adjacent sequences:  A270097 A270098 A270099 * A270101 A270102 A270103

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 11 2016

STATUS

approved

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Last modified May 7 16:00 EDT 2021. Contains 343652 sequences. (Running on oeis4.)