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A272453 Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 478", based on the 5-celled von Neumann neighborhood. 1
1, 6, 18, 39, 71, 115, 188, 264, 373, 477, 662, 818, 1055, 1271, 1636, 1888, 2285, 2653, 3194, 3586, 4235, 4783, 5584, 6188, 7005, 7817, 8814, 9690, 10807, 11843, 13196, 14304, 15845, 17213, 18874, 20454, 22239, 24079, 26160, 28060, 30337, 32409, 34862 (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=478; 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. A272451.

Sequence in context: A270215 A129863 A272014 * A271252 A271541 A035489

Adjacent sequences:  A272450 A272451 A272452 * A272454 A272455 A272456

KEYWORD

nonn,easy

AUTHOR

Robert Price, Apr 29 2016

STATUS

approved

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Last modified August 7 19:57 EDT 2020. Contains 336279 sequences. (Running on oeis4.)