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A272048 First differences of number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 419", based on the 5-celled von Neumann neighborhood. 1
4, 8, 15, -3, 76, -45, 113, -73, 169, -128, 204, -200, 412, -320, 504, -404, 524, -504, 704, -628, 924, -869, 1101, -969, 1225, -1148, 1368, -1308, 1796, -1660, 2028, -1784, 2008, -2004, 2360, -2180, 2700, -2648, 3184, -2892, 3108, -3112, 3620, -3424, 3976 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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..127

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=419; 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[on[[i+1]]-on[[i]], {i, 1, Length[on]-1}] (* Difference at each stage *)

CROSSREFS

Cf. A272045.

Sequence in context: A324585 A242519 A174554 * A112312 A076343 A335382

Adjacent sequences: A272045 A272046 A272047 * A272049 A272050 A272051

KEYWORD

sign,easy

AUTHOR

Robert Price, Apr 18 2016

STATUS

approved

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Last modified February 2 06:25 EST 2023. Contains 360000 sequences. (Running on oeis4.)