A266435 - OEIS

%I #29 Feb 16 2025 08:33:28

%S 1,111,0,1111111,0,11111111111,0,111111111111111,0,

%T 1111111111111111111,0,11111111111111111111111,0,

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

%N Binary representation of the n-th iteration of the "Rule 23" elementary cellular automaton starting with a single ON (black) cell.

%C Rules 31, 55, 63, 87, 95, 119 and 127 also generate this sequence.

%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

%H Robert Price, <a href="/A266435/b266435.txt">Table of n, a(n) for n = 0..500</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>

%H S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</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>

%F From _Colin Barker_, Dec 30 2015 and Apr 15 2019: (Start)

%F a(n) = ((-1)^n-1)*(1-10^(2*n+1))/18 for n>0.

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

%F G.f.: (1+111*x-10001*x^2+1000*x^3+10000*x^4) / ((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. - _Karl V. Keller, Jr._, Aug 11 2021

%t rule=23; 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 for n in range(33)]) # _Karl V. Keller, Jr._, Aug 11 2021

%Y Cf. A266434, A266436.

%K nonn,easy

%O 0,2

%A _Robert Price_, Dec 29 2015