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%I #24 Sep 14 2021 14:39:59
%S 1,110,111,1111010,11111,11111100010,1011111,111111111100010,1011111,
%T 1111111111111100010,1011111,11111111111111111100010,1011111,
%U 111111111111111111111100010,1011111,1111111111111111111111111100010,1011111,11111111111111111111111111111100010
%N Binary representation of the n-th iteration of the "Rule 111" elementary cellular automaton starting with a single ON (black) cell.
%H Robert Price, <a href="/A267254/b267254.txt">Table of n, a(n) for n = 0..1000</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_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,10001,0,-10000).
%F From _Colin Barker_, Jan 13 2016 and Apr 19 2019: (Start)
%F a(n) = 10001*a(n-2)-10000*a(n-4) for n>8.
%F G.f.: (1 +110*x -9890*x^2 +10900*x^3 -1089000*x^4 +989000*x^5 -109000000*x^6 +110000000*x^7 -10000000000*x^8) / ((1 -x)*(1 +x)*(1 -100*x)*(1 +100*x)).
%F (End)
%F a(n) = floor(10*100^n/9) - 11101 for odd n>4; a(n) = 1011111 for even n>4. - _Karl V. Keller, Jr._, Sep 14 2021
%t rule=111; 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]]],{k,1,rows}] (* Binary Representation of Rows *)
%o (Python) print([1, 110, 111, 1111010, 11111] + [10*100**n//9 - 11101 if n%2 else 1011111 for n in range(5,50)]) # _Karl V. Keller, Jr._, Sep 14 2021
%Y Cf. A267253, A267255.
%K nonn,easy
%O 0,2
%A _Robert Price_, Jan 12 2016
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