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%I #19 Aug 28 2021 03:20:19
%S 1,2,5,106,21,1962,85,32426,341,522922,1365,8383146,5461,134195882,
%T 21845,2147396266,87381,34359388842,349525,549754415786,1398101,
%U 8796087429802,5592405,140737465985706,22369621,2251799724206762,89478485,36028796661050026,357913941
%N Decimal representation of the n-th iteration of the "Rule 13" elementary cellular automaton starting with a single ON (black) cell.
%D Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H Robert Price, <a href="/A266284/b266284.txt">Table of n, a(n) for n = 0..999</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,21,0,-84,0,64).
%F From _Colin Barker_, Dec 27 2015 and Apr 14 2019: (Start)
%F a(n) = 21*a(n-2)-84*a(n-4)+64*a(n-6) for n>5.
%F G.f.: (1+2*x-16*x^2+64*x^3-96*x^5) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)*(1-4*x)*(1+4*x)).
%F (End)
%F a(n) = 2*4^n - (8*2^n-1)/3 - 1 for odd n; a(n) = (4*2^n-1)/3 for even. - _Karl V. Keller, Jr._, Aug 27 2021
%t rule=13; 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[FromDigits[catri[[k]],2],{k,1,rows}] (* Decimal Representation of Rows *)
%o (Python) print([2*4**n - (8*2**n-1)//3 - 1 if n%2 else (4*2**n-1)//3 for n in range(50)]) # _Karl V. Keller, Jr._, Aug 27 2021
%Y Cf. A266282, A266283.
%K nonn,easy
%O 0,2
%A _Robert Price_, Dec 26 2015
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