cp's OEIS Frontend

A290113 Decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood.

%I A290113 #22 Feb 16 2025 08:33:49
%S A290113 1,3,5,13,29,61,125,253,509,1021,2045,4093,8189,16381,32765,65533,
%T A290113 131069,262141,524285,1048573,2097149,4194301,8388605,16777213,
%U A290113 33554429,67108861,134217725,268435453,536870909,1073741821,2147483645,4294967293,8589934589
%N A290113 Decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood.
%C A290113 Initialized with a single black (ON) cell at stage zero.
%D A290113 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A290113 Robert Price, <a href="/A290113/b290113.txt">Table of n, a(n) for n = 0..126</a>
%H A290113 Robert Price, <a href="/A290113/a290113.tmp.txt">Diagrams of first 20 stages</a>
%H A290113 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 A290113 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A290113 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A290113 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A290113 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A290113 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A290113 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H A290113 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).
%F A290113 For n>1, a(n) = 2^(n+1)-3.
%F A290113 a(n) = A036563(n+1) for n > 1. - _Georg Fischer_, Oct 30 2018
%F A290113 From _Chai Wah Wu_, Apr 02 2024: (Start)
%F A290113 a(n) = 3*a(n-1) - 2*a(n-2) for n > 3.
%F A290113 G.f.: (4*x^3 - 2*x^2 + 1)/(2*x^2 - 3*x + 1). (End)
%t A290113 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A290113 code = 643; stages = 128;
%t A290113 rule = IntegerDigits[code, 2, 10];
%t A290113 g = 2 * stages + 1; (* Maximum size of grid *)
%t A290113 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A290113 ca = a;
%t A290113 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A290113 PrependTo[ca, a];
%t A290113 (* Trim full grid to reflect growth by one cell at each stage *)
%t A290113 k = (Length[ca[[1]]] + 1)/2;
%t A290113 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 A290113 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]
%Y A290113 Essentially the same as A091270.
%Y A290113 Cf. A036563, A290111, A290112, A290114.
%K A290113 nonn,easy
%O A290113 0,2
%A A290113 _Robert Price_, Jul 19 2017