A285646 - OEIS

%I #10 Apr 25 2017 09:59:09

%S 1,1,4,3,16,15,96,63,384,255,1536,1023,6144,4095,24576,16383,98304,

%T 65535,393216,262143,1572864,1048575,6291456,4194303,25165824,

%U 16777215,100663296,67108863,402653184,268435455,1610612736,1073741823,6442450944,4294967295

%N Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 65", based on the 5-celled von Neumann neighborhood.

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

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

%H Robert Price, <a href="/A285646/b285646.txt">Table of n, a(n) for n = 0..126</a>

%H Robert Price, <a href="/A285646/a285646.tmp.txt">Diagrams of first 20 stages</a>

%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 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</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_, Apr 24 2017: (Start)

%F G.f.: (1 + x - x^2 - 2*x^3 + 4*x^5 + 32*x^6 - 32*x^8) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).

%F a(n) = 3*2^(n-1) for n>4 and even.

%F a(n) = 2^(n-1) - 1 for n>4 and odd.

%F a(n) = 5*a(n-2) - 4*a(n-4) for n>3.

%F (End)

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

%t code = 65; 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 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

%Y Cf. A285643, A285644, A285645.

%K nonn,easy

%O 0,3

%A _Robert Price_, Apr 23 2017