A273910 Binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 614", based on the 5-celled von Neumann neighborhood.

1, 11, 101, 1011, 10001, 110111, 1000101, 11011011, 100000001, 1100000111, 10100010101, 101101101011, 1000000010001, 11000001110111, 101000101000101, 1011011011011011, 10000000000000001, 110000000000000111, 1010000000000010101, 10110000000001101011

Offset: 0

Author

Robert Price, Jun 03 2016

Keywords

• nonn
• easy

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..126
• Robert Price, Diagrams of the first 20 stages
• 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

Programs

Mathematica

CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=614; 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}];
Table[FromDigits[Part[ca[[i]][[i]],Range[i,2*i-1]],10], {i,1,stages-1}]