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A266722 Number of ON (black) cells in the n-th iteration of the "Rule 59" elementary cellular automaton starting with a single ON (black) cell.

%I A266722 #30 Feb 16 2025 08:33:28
%S A266722 1,2,2,6,2,10,2,14,2,18,2,22,2,26,2,30,2,34,2,38,2,42,2,46,2,50,2,54,
%T A266722 2,58,2,62,2,66,2,70,2,74,2,78,2,82,2,86,2,90,2,94,2,98,2,102,2,106,2,
%U A266722 110,2,114,2,118,2,122,2,126,2,130,2,134,2,138,2
%N A266722 Number of ON (black) cells in the n-th iteration of the "Rule 59" elementary cellular automaton starting with a single ON (black) cell.
%C A266722 For n >= 3, also the number of maximum independent vertex sets in the n-prism graph. - _Eric W. Weisstein_, Mar 30 2017
%C A266722 For n >= 3, also the number of maximum independent edge sets in the n-web graph. - _Eric W. Weisstein_, Dec 31 2017
%C A266722 For n >= 10, also the number of minimum dominating sets in the n-gear graph. - _Eric W. Weisstein_, Sep 09 2021
%H A266722 Robert Price, <a href="/A266722/b266722.txt">Table of n, a(n) for n = 0..1000</a>
%H A266722 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A266722 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GearGraph.html">Gear Graph</a>
%H A266722 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Matching.html">Matching</a>
%H A266722 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximumIndependentEdgeSet.html">Maximum Independent Edge Set</a>
%H A266722 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximumIndependentVertexSet.html">Maximum Independent Vertex Set</a>
%H A266722 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimumDominatingSet.html">Minimum Dominating Set</a>
%H A266722 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrismGraph.html">Prism Graph</a>
%H A266722 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WebGraph.html">Web Graph</a>
%H A266722 Stephen Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>, Wolfram Media, 2002; p. 55.
%H A266722 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A266722 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H A266722 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A266722 From _Colin Barker_, Jan 05 2016 and Apr 17 2019: (Start)
%F A266722 a(n) = 1+(-1)^n+n-(-1)^n*n for n>0.
%F A266722 a(n) = 2*a(n-2)-a(n-4) for n>4.
%F A266722 G.f.: (1+2*x-x^2)*(1+x^2) / ((1-x)^2*(1+x)^2). (End)
%t A266722 rule=59; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[Total[catri[[k]]],{k,1,rows}] (* Number of Black cells in stage n *)
%t A266722 Join[{1}, Table[Piecewise[{{2, Mod[n, 2] == 0}, {2 n, Mod[n, 2] == 1}}], {n, 20}]] (* _Eric W. Weisstein_, Sep 09 2021 *)
%t A266722 Join[{1}, LinearRecurrence[{0, 2, 0, -1}, {2, 2, 6, 2}, 20]] (* _Eric W. Weisstein_, Sep 09 2021 *)
%t A266722 CoefficientList[Series[(1 + 2 x + 2 x^3 - x^4)/(-1 + x^2)^2, {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 09 2021 *)
%Y A266722 Cf. A266716, A266723 (partial sums), A266724.
%K A266722 nonn,easy
%O A266722 0,2
%A A266722 _Robert Price_, Jan 03 2016