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A270093 Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 78", based on the 5-celled von Neumann neighborhood. 1
1, 6, 14, 34, 54, 86, 134, 190, 278, 334, 430, 514, 654, 782, 962, 1234, 1494, 1670, 1982, 2222, 2550, 2830, 3238, 3638, 4190, 4498, 5034, 5510, 6258, 6818, 7666, 8450, 9514, 10282, 11374, 12202, 13246, 14174, 15242, 16438, 17538, 18662, 19994, 21402, 23062 (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=78; 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. A270091.

Sequence in context: A269717 A271097 A269709 * A270735 A270170 A270986

Adjacent sequences:  A270090 A270091 A270092 * A270094 A270095 A270096

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 10 2016

STATUS

approved

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Last modified April 20 09:21 EDT 2021. Contains 343125 sequences. (Running on oeis4.)