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%I #35 Aug 31 2021 20:03:11
%S 1,100,10,1111001,100,11111110011,1000,111111111100111,10000,
%T 1111111111111001111,100000,11111111111111110011111,1000000,
%U 111111111111111111100111111,10000000,1111111111111111111111001111111,100000000,11111111111111111111111110011111111
%N Binary representation of the n-th iteration of the "Rule 3" elementary cellular automaton starting with a single ON (black) cell.
%C Rule 35 also generates this sequence.
%H Robert Price, <a href="/A266068/b266068.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 Stephen Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>, Wolfram Media, 2002; p. 55.
%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 (0,10011,0,-110010,0,100000).
%F G.f.: (1+100*x-10001*x^2+109901*x^3+10000*x^4-120000*x^5) / ((1-x)*(1+x)*(1-100*x)*(1+100*x)*(1-10*x^2)). - _Colin Barker_, Dec 21 2015
%F a(n) = (10*100^n - 99*10^((n-1)/2) - 1)/9 for odd n; a(n) = 10^(n/2) for even n. - _Karl V. Keller, Jr._, Aug 26 2021
%e From _Michael De Vlieger_, Dec 21 2015: (Start)
%e First 8 rows, replacing leading zeros with ".", the row converted to its binary equivalent at right:
%e 1 = 1
%e 1 0 0 = 100
%e . . . 1 0 = 10
%e 1 1 1 1 0 0 1 = 1111001
%e . . . . . . 1 0 0 = 100
%e 1 1 1 1 1 1 1 0 0 1 1 = 11111110011
%e . . . . . . . . . 1 0 0 0 = 1000
%e 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 = 111111111100111
%e (End)
%t rule = 3; rows = 20; Table[FromDigits[Table[Take[CellularAutomaton[rule,{{1},0}, rows-1, {All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]]], {k,1,rows}]
%o (Python) print([(10*100**n - 99*10**((n-1)//2) - 1)//9 if n%2 else 10**(n//2) for n in range(30)]) # _Karl V. Keller, Jr._, Aug 26 2021
%Y Cf. A263428, A266069.
%K nonn,easy
%O 0,2
%A _Robert Price_, Dec 20 2015
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