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%I A271051 #16 Jul 26 2024 21:16:35
%S A271051 1,8,4,44,13,116,13,220,13,356,13,524,13,724,13,956,13,1220,13,1516,
%T A271051 13,1844,13,2204,13,2596,13,3020,13,3476,13,3964,13,4484,13,5036,13,
%U A271051 5620,13,6236,13,6884,13,7564,13,8276,13,9020,13,9796,13,10604,13,11444
%N A271051 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 253", based on the 5-celled von Neumann neighborhood.
%C A271051 Initialized with a single black (ON) cell at stage zero.
%D A271051 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A271051 Robert Price, <a href="/A271051/b271051.txt">Table of n, a(n) for n = 0..128</a>
%H A271051 Robert Price, <a href="/A271051/a271051.tmp.txt">Diagrams of the first 20 stages.</a>
%H A271051 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 A271051 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A271051 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A271051 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A271051 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A271051 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A271051 Conjectures from _Colin Barker_, Nov 22 2017: (Start)
%F A271051 G.f.: (1 + 8*x + x^2 + 20*x^3 + 4*x^4 + 8*x^5 - 15*x^6 - 4*x^7 + 9*x^8) / ((1 - x)^3*(1 + x)^3).
%F A271051 a(n) = 13 for n>2 and even.
%F A271051 a(n) = 4*(n^2 + n - 1) for n>2 and odd.
%F A271051 a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>8.
%F A271051 (End)
%t A271051 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A271051 code=253; stages=128;
%t A271051 rule=IntegerDigits[code,2,10];
%t A271051 g=2*stages+1; (* Maximum size of grid *)
%t A271051 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A271051 ca=a;
%t A271051 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A271051 PrependTo[ca,a];
%t A271051 (* Trim full grid to reflect growth by one cell at each stage *)
%t A271051 k=(Length[ca[[1]]]+1)/2;
%t A271051 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 A271051 Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%K A271051 nonn,easy
%O A271051 0,2
%A A271051 _Robert Price_, Mar 29 2016

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