A270222 Number of active (ON,black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 131", based on the 5-celled von Neumann neighborhood.

1, 5, 33, 161, 705, 2945, 12033, 48641, 195585, 784385, 3141633, 12574721, 50315265, 201293825, 805240833, 3221094401

Offset

0,2

Comments

Initialized with a single black (ON) cell at stage zero.

It is conjectured that Rules 163, 171, 227, 235, 771, 787, 803, 811, 819, 827, 835, 851, 867, 875, 883 and 891 also generate this sequence. - Lars Blomberg, Apr 30 2016

Also the number of vertex cuts in the (n+1)-barbell graph for n > 1. - Eric W. Weisstein, Apr 23 2023

References

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

Table of n, a(n) for n=0..15.

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, Barbell Graph

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

Eric Weisstein's World of Mathematics, Vertex Cut

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, Mar 13 2016: (Start)

a(n) = 1-2^(2+n)+3*4^n.

a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>3.

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

(End)

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=131; 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}];
on=Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)
Part[on, 2^Range[0, Log[2, stages]]] (* Extract relevant terms *)

Crossrefs

Cf. A270221.

Sequence in context: A146263 A359703 A255943 * A270279 A270690 A273141

Adjacent sequences: A270219 A270220 A270221 * A270223 A270224 A270225

Keyword

nonn,more

Author

Robert Price, Mar 13 2016

Extensions

a(8)-a(15) from Lars Blomberg, Apr 30 2016

Status

approved