cp's OEIS Frontend

A266068 Binary representation of the n-th iteration of the "Rule 3" elementary cellular automaton starting with a single ON (black) cell.

%I A266068 #36 Feb 16 2025 08:33:28
%S A266068 1,100,10,1111001,100,11111110011,1000,111111111100111,10000,
%T A266068 1111111111111001111,100000,11111111111111110011111,1000000,
%U A266068 111111111111111111100111111,10000000,1111111111111111111111001111111,100000000,11111111111111111111111110011111111
%N A266068 Binary representation of the n-th iteration of the "Rule 3" elementary cellular automaton starting with a single ON (black) cell.
%C A266068 Rule 35 also generates this sequence.
%H A266068 Robert Price, <a href="/A266068/b266068.txt">Table of n, a(n) for n = 0..999</a>
%H A266068 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A266068 Stephen Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>, Wolfram Media, 2002; p. 55.
%H A266068 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A266068 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H A266068 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,10011,0,-110010,0,100000).
%F A266068 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 A266068 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 A266068 From _Michael De Vlieger_, Dec 21 2015: (Start)
%e A266068 First 8 rows, replacing leading zeros with ".", the row converted to its binary equivalent at right:
%e A266068               1                =               1
%e A266068             1 0 0              =             100
%e A266068           . . . 1 0            =              10
%e A266068         1 1 1 1 0 0 1          =         1111001
%e A266068       . . . . . . 1 0 0        =             100
%e A266068     1 1 1 1 1 1 1 0 0 1 1      =     11111110011
%e A266068   . . . . . . . . . 1 0 0 0    =            1000
%e A266068 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1  = 111111111100111
%e A266068 (End)
%t A266068 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 A266068 (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 A266068 Cf. A263428, A266069.
%K A266068 nonn,easy
%O A266068 0,2
%A A266068 _Robert Price_, Dec 20 2015