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A284485 Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 961", based on the 5-celled von Neumann neighborhood.

%I A284485 #11 Feb 16 2025 08:33:43
%S A284485 1,0,7,11,15,47,63,191,255,767,1023,3071,4095,12287,16383,49151,65535,
%T A284485 196607,262143,786431,1048575,3145727,4194303,12582911,16777215,
%U A284485 50331647,67108863,201326591,268435455,805306367,1073741823,3221225471,4294967295,12884901887
%N A284485 Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 961", based on the 5-celled von Neumann neighborhood.
%C A284485 Initialized with a single black (ON) cell at stage zero.
%D A284485 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A284485 Robert Price, <a href="/A284485/b284485.txt">Table of n, a(n) for n = 0..126</a>
%H A284485 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 A284485 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A284485 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A284485 Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>
%H A284485 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A284485 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A284485 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H A284485 Robert Price, <a href="/A284485/a284485.tmp.txt">Diagrams of first 20 stages</a>
%F A284485 Conjectures from _Colin Barker_, Mar 28 2017: (Start)
%F A284485 G.f.: (1 - x + 3*x^2 + 8*x^3 - 24*x^4 + 16*x^5) / ((1 - x)*(1 - 2*x)*(1 + 2*x)).
%F A284485 a(n) = (-4 - (-2)^n + 5*2^n)/4 for n>2.
%F A284485 a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) for n>5.
%F A284485 (End)
%t A284485 CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
%t A284485 code = 961; stages = 128;
%t A284485 rule = IntegerDigits[code, 2, 10];
%t A284485 g = 2 * stages + 1; (* Maximum size of grid *)
%t A284485 a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
%t A284485 ca = a;
%t A284485 ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
%t A284485 PrependTo[ca, a];
%t A284485 (* Trim full grid to reflect growth by one cell at each stage *)
%t A284485 k = (Length[ca[[1]]] + 1)/2;
%t A284485 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 A284485 Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 2], {i ,1, stages - 1}]
%Y A284485 Cf. A284483, A284484, A101622.
%K A284485 nonn,easy
%O A284485 0,3
%A A284485 _Robert Price_, Mar 27 2017