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A265720 Binary representation of the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.

%I A265720 #37 Feb 16 2025 08:33:28
%S A265720 1,0,100,1100011,10000,11110001111,1000000,111111000111111,100000000,
%T A265720 1111111100011111111,10000000000,11111111110001111111111,
%U A265720 1000000000000,111111111111000111111111111,100000000000000,1111111111111100011111111111111,10000000000000000
%N A265720 Binary representation of the n-th iteration of the "Rule 1" elementary cellular automaton starting with a single ON (black) cell.
%C A265720 Rule 33 also generates this sequence.
%H A265720 Robert Price, <a href="/A265720/b265720.txt">Table of n, a(n) for n = 0..999</a>
%H A265720 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A265720 Stephen Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>, Wolfram Media, 2002; p. 55.
%H A265720 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A265720 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H A265720 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,10101,0,-1010100,0,1000000).
%F A265720 From _Colin Barker_, Dec 14 2015 and Apr 16 2019: (Start)
%F A265720 a(n) = 10101*a(n-2) - 1010100*a(n-4) + 1000000*a(n-6) for n > 5.
%F A265720 G.f.: (1-10001*x^2+1100011*x^3+10000*x^4-1210000*x^5) / ((1-x)*(1+x)*(1-10*x)*(1+10*x)*(1-100*x)*(1+100*x)).
%F A265720 (End)
%F A265720 a(n) = (10*100^n - 999*10^(n-1) - 1)/9 for odd n; a(n) = 10^n for even n. - _Karl V. Keller, Jr._, Aug 25 2021
%e A265720 From _Michael De Vlieger_, Dec 14 2015: (Start)
%e A265720 First 10 rows, replacing leading zeros with ".", the row converted to its binary equivalent at right:
%e A265720                   1                    =                   1
%e A265720                 . . 0                  =                   0
%e A265720               . . 1 0 0                =                 100
%e A265720             1 1 0 0 0 1 1              =             1100011
%e A265720           . . . . 1 0 0 0 0            =               10000
%e A265720         1 1 1 1 0 0 0 1 1 1 1          =         11110001111
%e A265720       . . . . . . 1 0 0 0 0 0 0        =             1000000
%e A265720     1 1 1 1 1 1 0 0 0 1 1 1 1 1 1      =     111111000111111
%e A265720   . . . . . . . . 1 0 0 0 0 0 0 0 0    =           100000000
%e A265720 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1  = 1111111100011111111
%e A265720 (End)
%t A265720 rule = 1; 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 A265720 (Python) print([(10*100**n - 999*10**(n-1) - 1)//9 if n%2 else 10**n for n in range(50)]) # _Karl V. Keller, Jr._, Aug 25 2021
%Y A265720 Cf. A265718, A265721.
%K A265720 nonn,easy
%O A265720 0,3
%A A265720 _Robert Price_, Dec 14 2015