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A272258 Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 451", based on the 5-celled von Neumann neighborhood. 1
1, 6, 15, 39, 60, 132, 177, 289, 382, 606, 723, 1043, 1216, 1680, 1885, 2413, 2706, 3558, 3783, 4883, 5028, 6528, 6897, 8293, 8902, 10674, 11271, 13271, 13932, 16344, 17133, 19525, 20558, 23594, 24439, 28084, 28769, 32817, 33974, 38242, 39427, 44339, 45584 (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=451; 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. A272256.

Sequence in context: A254008 A277089 A271545 * A192308 A277237 A171159

Adjacent sequences:  A272255 A272256 A272257 * A272259 A272260 A272261

KEYWORD

nonn,easy

AUTHOR

Robert Price, Apr 23 2016

STATUS

approved

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Last modified October 17 08:38 EDT 2018. Contains 316276 sequences. (Running on oeis4.)