A282485 Decimal 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 478", based on the 5-celled von Neumann neighborhood.

1, 3, 3, 15, 3, 31, 99, 127, 387, 479, 1763, 1919, 7043, 7647, 28387, 30591, 113539, 122335, 454371, 489343, 1817475, 1957343, 7270115, 7829375, 29080451, 31317471, 116322019, 125269887, 465288067, 501079519, 1861152483, 2004318079, 7444609923, 8017272287

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

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, Feb 18 2017: (Start)

a(n) = 4*a(n-2) + a(n-4) - 4*a(n-6) for n>6.

G.f.: (1 + 3*x - x^2 + 3*x^3 - 10*x^4 - 32*x^5 + 88*x^6 + 128*x^10) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)*(1 + x^2)).

(End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 478; 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]], 2], {i , 1, stages - 1}]

Crossrefs

Cf. A282482, A282483, A282484.

Sequence in context: A260119 A368117 A282009 * A100371 A286206 A286501

Adjacent sequences: A282482 A282483 A282484 * A282486 A282487 A282488

Keyword

nonn,easy

Author

Robert Price, Feb 16 2017

Status

approved