A278666 - OEIS

%I #9 Nov 27 2016 16:02:58

%S 1,3,6,12,24,50,102,196,388,804,1652,3156,6228,12884,26452,50516,

%T 99668,206164,423252,808276,1594708,3298644,6772052,12932436,25515348,

%U 52778324,108352852,206918996,408245588,844453204,1733645652,3310703956,6531929428

%N Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 62", 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="/A278666/b278666.txt">Table of n, a(n) for n = 0..126</a>

%H Robert Price, <a href="/A278666/a278666.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 <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_, Nov 26 2016: (Start)

%F a(n) = a(n-1) + 16*a(n-4) - 16*a(n-5) for n>7.

%F G.f.: (1 +2*x +3*x^2 +6*x^3 -4*x^4 -6*x^5 +4*x^6 -2*x^7 +16*x^10) / ((1 -x)*(1 -2*x)*(1 +2*x)*(1 +4*x^2)).

%F (End)

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

%t code=62; 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[1,i]],2], {i,1,stages-1}]

%Y Cf. A278664, A278665, A278667.

%K nonn,easy

%O 0,2

%A _Robert Price_, Nov 25 2016