A278900 Decimal representation of the x-axis, from the left edge to the origin, 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, 1, 1, 15, 4, 55, 0, 247, 20, 983, 20, 4087, 84, 16215, 84, 65495, 340, 261463, 340, 1048407, 1364, 4191575, 1364, 16776535, 5460, 67097943, 5460, 268432727, 21844, 1073698135, 21844, 4294956375, 87380, 17179694423, 87380, 68719433047, 349524, 274877207895

Offset

0,4

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

Wolfram Research, Wolfram Atlas of Simple Programs.

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) = 5*a(n-2) - 20*a(n-6) + 16*a(n-8) for n>15.

G.f.: (1 +x -4*x^2 +10*x^3 -x^4 -20*x^5 -8*x^7 +24*x^8 +32*x^9 -16*x^10 +32*x^11 -80*x^12 -160*x^13 +64*x^14 +128*x^15) / ((1 -x)*(1 +x)*(1 -2*x)*(1 +2*x)*(1 -2*x^2)*(1 +2*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, k}];
Table[FromDigits[Part[ca[[i]][[i]], Range[1, i]], 2], {i, 1, stages - 1}]

Crossrefs

Cf. A278898, A278899, A278901.

Sequence in context: A349710 A348924 A278872 * A278917 A279023 A272682

Adjacent sequences: A278897 A278898 A278899 * A278901 A278902 A278903

Keyword

nonn,easy

Author

Robert Price, Nov 30 2016

Status

approved