A285834 Decimal representation of the diagonal from the corner 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, 3, 3, 15, 6, 63, 12, 255, 30, 1023, 62, 4095, 126, 16383, 254, 65535, 510, 262143, 1022, 1048575, 2046, 4194303, 4094, 16777215, 8190, 67108863, 16382, 268435455, 32766, 1073741823, 65534, 4294967295, 131070, 17179869183, 262142, 68719476735, 524286

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, Apr 28 2017: (Start)

G.f.: (1+3*x-4*x^2-6*x^3-x^4+4*x^6+6*x^8-28*x^10+16*x^12) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)*(1-2*x^2)).

a(n) = 7*a(n-2) - 14*a(n-4) + 8*a(n-6) for n>6.

(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}]
LinearRecurrence[{0, 7, 0, -14, 0, 8}, {1, 3, 3, 15, 6, 63, 12, 255, 30, 1023, 62, 4095, 126}, 40] (* Harvey P. Dale, Feb 28 2023 *)

Crossrefs

Cf. A284940, A285833, A285835.

Sequence in context: A285947 A285843 A285822 * A285562 A285542 A367774

Adjacent sequences: A285831 A285832 A285833 * A285835 A285836 A285837

Keyword

nonn,easy

Author

Robert Price, Apr 27 2017

Status

approved