The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A270682 Number of active (ON,black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 190", based on the 5-celled von Neumann neighborhood. 0
 1, 5, 24, 88, 312, 1144, 4344, 16888, 66552, 264184, 1052664, 4202488, 16793592, 67141624, 268500984, 1073872888 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Initialized with a single black (ON) cell at stage zero. Conjecture: Rule 254 also generates this sequence. - Lars Blomberg, Jun 03 2016 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, Elementary Cellular Automaton 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 Conjecture: a(n) = 4^n + 4*2^n - 8, n>1. - Lars Blomberg, Jun 03 2016 Conjectures from Colin Barker, Jun 08 2016: (Start) a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>4. G.f.: (1-2*x+3*x^2-18*x^3-8*x^4) / ((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=190; 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. A270681. Sequence in context: A270906 A268587 A334459 * A272420 A089095 A220316 Adjacent sequences: A270679 A270680 A270681 * A270683 A270684 A270685 KEYWORD nonn,more AUTHOR Robert Price, Mar 21 2016 EXTENSIONS a(8)-a(15) from Lars Blomberg, Jun 03 2016 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 24 22:33 EDT 2024. Contains 373690 sequences. (Running on oeis4.)