cp's OEIS Frontend

A285613 Binary 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 62", based on the 5-celled von Neumann neighborhood.

%I A285613 #12 Feb 16 2025 08:33:44
%S A285613 1,10,10,100,1100,11000,111000,1110000,11110000,111100000,1111100000,
%T A285613 11111000000,111111000000,1111110000000,11111110000000,
%U A285613 111111100000000,1111111100000000,11111111000000000,111111111000000000,1111111110000000000,11111111110000000000
%N A285613 Binary 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 62", based on the 5-celled von Neumann neighborhood.
%C A285613 Initialized with a single black (ON) cell at stage zero.
%D A285613 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A285613 Robert Price, <a href="/A285613/b285613.txt">Table of n, a(n) for n = 0..126</a>
%H A285613 Robert Price, <a href="/A285613/a285613.tmp.txt">Diagrams of first 20 stages</a>
%H A285613 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 A285613 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A285613 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A285613 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A285613 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A285613 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A285613 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A285613 Conjectures from _Colin Barker_, Apr 23 2017: (Start)
%F A285613 G.f.: (1 - 100*x^2 + 1000*x^4) / ((1 - 10*x)*(1 - 10*x^2)).
%F A285613 a(n) = (10^n - 10^(n/2)) / 9 for n>1 and even.
%F A285613 a(n) = (10^n - 10^(n/2+1/2)) / 9 for n>1 and odd.
%F A285613 a(n) = 10*a(n-1) + 10*a(n-2) - 100*a(n-3) for n>2.
%F A285613 (End)
%t A285613 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A285613 code = 62; stages = 128;
%t A285613 rule = IntegerDigits[code, 2, 10];
%t A285613 g = 2 * stages + 1; (* Maximum size of grid *)
%t A285613 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A285613 ca = a;
%t A285613 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A285613 PrependTo[ca, a];
%t A285613 (* Trim full grid to reflect growth by one cell at each stage *)
%t A285613 k = (Length[ca[[1]]] + 1)/2;
%t A285613 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 A285613 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
%Y A285613 Cf. A285612, A056453, A233411.
%K A285613 nonn,easy
%O A285613 0,2
%A A285613 _Robert Price_, Apr 22 2017