A271286 - OEIS

%I #22 Jul 26 2024 21:16:38

%S 1,4,40,200,872,3624,14760,59560,239272,959144,3840680,15370920,

%T 61500072,246033064,984197800,3936922280

%N Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 337", based on the 5-celled von Neumann neighborhood.

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

%C It is conjectured that Rule 505 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 Conjectures from _Colin Barker_, May 04 2016: (Start)

%F a(n) = (11*4^n)/3-8/3-2^(2+n) for n>0.

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

%F G.f.: (1-3*x+26*x^2-32*x^3) / ((1-x)*(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=337; 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. A271285.

%K nonn,more

%O 0,2

%A _Robert Price_, Apr 03 2016

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