A278899 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 107", based on the 5-celled von Neumann neighborhood.

1, 10, 100, 1111, 100, 111011, 0, 11101111, 1010000, 1110101111, 101000000, 111011111111, 10101000000, 11101010111111, 1010100000000, 1110101111111111, 101010100000000, 111010101011111111, 10101010000000000, 11101010111111111111, 1010101010000000000

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..126

Robert Price, Diagrams of 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

Formula

Conjectures from Colin Barker, Nov 30 2016: (Start)

a(n) = 101*a(n-2) - 10100*a(n-6) + 10000*a(n-8) for n>15.

G.f.: (1 +10*x -x^2 +101*x^3 -10000*x^4 -1200*x^5 -10000*x^7 +2010000*x^8 +10000*x^9 -1000000*x^10 +1000000*x^11 -101000000*x^12 -101000000*x^13 +100000000*x^14 +100000000*x^15) / ((1 -x)*(1 +x)*(1 -10*x)*(1 +10*x)*(1 -10*x^2)*(1 +10*x^2)).

(End)

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=107; 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}]

Crossrefs

Cf. A278898, A278900, A278901.

Sequence in context: A283377 A283352 A278871 * A278916 A279016 A366647

Adjacent sequences: A278896 A278897 A278898 * A278900 A278901 A278902

Keyword

nonn,easy

Author

Robert Price, Nov 30 2016

Status

approved