cp's OEIS Frontend

A285475 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 3", based on the 5-celled von Neumann neighborhood.

%I A285475 #22 Feb 16 2025 08:33:44
%S A285475 1,3,4,15,16,63,64,255,256,1023,1024,4095,4096,16383,16384,65535,
%T A285475 65536,262143,262144,1048575,1048576,4194303,4194304,16777215,
%U A285475 16777216,67108863,67108864,268435455,268435456,1073741823,1073741824,4294967295,4294967296
%N A285475 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 3", based on the 5-celled von Neumann neighborhood.
%C A285475 Initialized with a single black (ON) cell at stage zero.
%D A285475 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A285475 Robert Price, <a href="/A285475/b285475.txt">Table of n, a(n) for n = 0..126</a>
%H A285475 Robert Price, <a href="/A285475/a285475.tmp.txt">Diagrams of first 20 stages</a>
%H A285475 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 A285475 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A285475 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A285475 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A285475 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A285475 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A285475 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H A285475 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,5,0,-4).
%F A285475 From _Colin Barker_, Apr 19 2017: (Start)
%F A285475 G.f.: (1 + 3*x - x^2) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
%F A285475 a(n) = (-1 - (-2)^n + (-1)^n + 3*2^n)/2.
%F A285475 a(n) = 5*a(n-2) - 4*a(n-4) for n>3. (End)
%F A285475 a(2*n-1) + a(2*n) = A083420(n). - _Paul Curtz_, Dec 16 2024
%t A285475 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A285475 code = 3; stages = 128;
%t A285475 rule = IntegerDigits[code, 2, 10];
%t A285475 g = 2 * stages + 1; (* Maximum size of grid *)
%t A285475 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A285475 ca = a;
%t A285475 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A285475 PrependTo[ca, a];
%t A285475 (* Trim full grid to reflect growth by one cell at each stage *)
%t A285475 k = (Length[ca[[1]]] + 1)/2;
%t A285475 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 A285475 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
%Y A285475 Cf. A285473, A285474, A080924.
%Y A285475 Cf. A083420.
%Y A285475 Cf. A000302 (even bisection), A024036 (odd bisection).
%K A285475 nonn,easy
%O A285475 0,2
%A A285475 _Robert Price_, Apr 19 2017