

A270171


First differences of number of active (ON,black) cells in nth stage of growth of twodimensional cellular automaton defined by "Rule 110", based on the 5celled von Neumann neighborhood.


1



4, 3, 12, 0, 12, 16, 20, 8, 4, 16, 20, 24, 36, 40, 52, 0, 20, 16, 12, 40, 20, 56, 36, 32, 36, 88, 12, 136, 68, 128, 100, 16, 100, 8, 28, 88, 20, 72, 36, 24, 60, 88, 52, 136, 108, 96, 92, 56, 84, 80, 12, 136, 220, 8, 212, 16, 188, 248, 44, 352, 356
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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 5Neighbor 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=110; 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+1n, k1+n]], {j, k+1n, k1+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. A270168.
Sequence in context: A169704 A270094 A270736 * A270987 A204291 A270324
Adjacent sequences: A270168 A270169 A270170 * A270172 A270173 A270174


KEYWORD

sign,easy


AUTHOR

Robert Price, Mar 12 2016


STATUS

approved



