A271061 - OEIS

%I #15 Jul 26 2024 21:16:36

%S 1,8,48,224,960,3968,16128,65024,261120,1046528,4190208,16769024,

%T 67092480,268402688,1073676288,4294836224

%N Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 261", based on the 5-celled von Neumann neighborhood.

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

%C Similar to A211012.

%C It is conjectured that Rules 269, 277, 285, 293, 301, 309, 317, 325, 333, 341, 349, 357, 365, 373, 381, 413, 445, 477, 509, 645, 653, 661, 669, 677, 685, 693, 701, 709, 717, 725, 733, 741, 749, 757 and 765 also produces this sequence.  It would be nice to have a proof.

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

%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>

%H S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%H <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>

%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>

%F Conjecture: a(n) = 4*4^n - 4*2^n, n>0. - _Lars Blomberg_, Jun 09 2016

%F Conjectures from _Colin Barker_, Dec 01 2016: (Start)

%F a(n) = 6*a(n-1) - 8*a(n-2) for n>2.

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

%F (End)

%t CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];

%t code=261; stages=128;

%t rule=IntegerDigits[code,2,10];

%t g=2*stages+1; (* Maximum size of grid *)

%t a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)

%t ca=a;

%t ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];

%t PrependTo[ca,a];

%t (* Trim full grid to reflect growth by one cell at each stage *)

%t k=(Length[ca[[1]]]+1)/2;

%t ca=Table[Table[Part[ca[[n]][[j]],Range[k+1-n,k-1+n]],{j,k+1-n,k-1+n}],{n,1,k}];

%t on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)

%t Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)

%Y Cf. A211012, A271060.

%K nonn,more

%O 0,2

%A _Robert Price_, Mar 29 2016

%E a(8)-a(15) from _Lars Blomberg_, Jun 09 2016