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A266069 Decimal representation of the n-th iteration of the "Rule 3" elementary cellular automaton starting with a single ON (black) cell.

%I A266069 #40 Feb 16 2025 08:33:28
%S A266069 1,4,2,121,4,2035,8,32743,16,524239,32,8388511,64,134217535,128,
%T A266069 2147483263,256,34359737599,512,549755812351,1024,8796093019135,2048,
%U A266069 140737488349183,4096,2251799813672959,8192,36028797018939391,16384,576460752303374335,32768
%N A266069 Decimal representation of the n-th iteration of the "Rule 3" elementary cellular automaton starting with a single ON (black) cell.
%C A266069 Rule 35 also generates this sequence.
%D A266069 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
%H A266069 Robert Price, <a href="/A266069/b266069.txt">Table of n, a(n) for n = 0..999</a>
%H A266069 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A266069 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A266069 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A266069 G.f.: (1+4*x-17*x^2+45*x^3+16*x^4-64*x^5) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)*(1-2*x^2)). - _Colin Barker_, Dec 21 2015 and Apr 18 2019
%F A266069 a(n) = 2*4^n - 3*2^((n-1)/2) - 1 for odd n; a(n) = 2^(n/2) for even n. - _Karl V. Keller, Jr._, Aug 25 2021
%e A266069 From _Michael De Vlieger_, Dec 21 2015: (Start)
%e A266069 First 8 rows, replacing leading zeros with ".", the row converted to its binary, then decimal equivalent at right:
%e A266069               1                =               1 ->     1
%e A266069             1 0 0              =             100 ->     4
%e A266069           . . . 1 0            =              10 ->     2
%e A266069         1 1 1 1 0 0 1          =         1111001 ->   121
%e A266069       . . . . . . 1 0 0        =             100 ->     4
%e A266069     1 1 1 1 1 1 1 0 0 1 1      =     11111110011 ->  2035
%e A266069   . . . . . . . . . 1 0 0 0    =            1000 ->     8
%e A266069 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1  = 111111111100111 -> 32743
%e A266069 (End)
%t A266069 rule = 3; rows = 30; Table[FromDigits[Table[Take[CellularAutomaton[rule,{{1},0}, rows-1, {All,All}][[k]], {rows-k+1, rows+k-1}], {k,1,rows}][[k]],2], {k,1,rows}]
%o A266069 (Python) print([2*4**n - 3*2**((n-1)//2) - 1 if n%2 else 2**(n//2) for n in range(30)]) # _Karl V. Keller, Jr._, Aug 25 2021
%Y A266069 Cf. A263428, A266068, A266070, A266071, A081253, A266072, A247375, A266073, A266074.
%K A266069 nonn,easy
%O A266069 0,2
%A A266069 _Robert Price_, Dec 20 2015