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
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
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
KEYWORD
nonn,more
AUTHOR
Robert Price, Mar 13 2016
EXTENSIONS
a(8)-a(15) from Lars Blomberg, Apr 30 2016
STATUS
approved