A266323 - OEIS

%I #23 Sep 03 2021 09:14:18

%S 1,101,0,1111111,0,11111111111,0,111111111111111,0,

%T 1111111111111111111,0,11111111111111111111111,0,

%U 111111111111111111111111111,0,1111111111111111111111111111111,0,11111111111111111111111111111111111,0,111111111111111111111111111111111111111

%N Binary representation of the n-th iteration of the "Rule 19" elementary cellular automaton starting with a single ON (black) cell.

%H Robert Price, <a href="/A266323/b266323.txt">Table of n, a(n) for n = 0..500</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_, Dec 28 2015 and Apr 15 2019: (Start)

%F a(n) = 10001*a(n-2) - 10000*a(n-4) for n>5.

%F G.f.: (1+101*x-10001*x^2+101010*x^3+10000*x^4-100000*x^5) / ((1-x)*(1+x)*(1-100*x)*(1+100*x)).

%F (End)

%F a(n) = (10*100^n - 1)/9*(n mod 2) + 0^n - 10*0^abs(n-1). - _Karl V. Keller, Jr._, Sep 02 2021

%t rule=19; 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([(10*100**n - 1)//9*(n%2) + 0**n - 10*0**abs(n-1) for n in range(50)]) # _Karl V. Keller, Jr._, Sep 02 2021

%Y Cf. A266155, A266324.

%K nonn,easy

%O 0,2

%A _Robert Price_, Dec 27 2015