A265380 - OEIS

%I #36 Jul 06 2023 12:30:28

%S 1,11,111,1110,11101,111011,1110111,11101110,111011101,1110111011,

%T 11101110111,111011101110,1110111011101,11101110111011,

%U 111011101110111,1110111011101110,11101110111011101,111011101110111011,1110111011101110111,11101110111011101110

%N Binary representation of the middle column of the "Rule 158" elementary cellular automaton starting with a single ON (black) cell.

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

%H Robert Price, <a href="/A265380/b265380.txt">Table of n, a(n) for n = 0..999</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</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 Conjectures from _Colin Barker_, Dec 14 2015 and Apr 18 2019: (Start)

%F a(n) = 10*a(n-1) + a(n-4) - 10*a(n-5) for n>4.

%F G.f.: (1+x+x^2) / ((1-x)*(1+x)*(1-10*x)*(1+x^2)).

%F (End)

%e From _Michael De Vlieger_, Dec 09 2015: (Start)

%e First 8 rows at left, ignoring "0" outside of range of 1's, the center column values in parentheses, and at right the value of center column cells up to that row :

%e                         (1)                          -> 1

%e                       1 (1) 1                        -> 11

%e                     1 1 (1) 0 1                      -> 111

%e                   1 1 1 (0) 0 1 1                    -> 1110

%e                 1 1 1 0 (1) 1 1 0 1                  -> 11101

%e               1 1 1 0 0 (1) 1 0 0 1 1                -> 111011

%e             1 1 1 0 1 1 (1) 0 1 1 1 0 1              -> 1110111

%e           1 1 1 0 0 1 1 (0) 0 1 1 0 0 1 1            -> 11101110

%e         1 1 1 0 1 1 1 0 (1) 1 1 0 1 1 1 0 1          -> 111011101

%e       1 1 1 0 0 1 1 0 0 (1) 1 0 0 1 1 0 0 1 1        -> 1110111011

%e     1 1 1 0 1 1 1 0 1 1 (1) 0 1 1 1 0 1 1 1 0 1      -> 11101110111

%e   1 1 1 0 0 1 1 0 0 1 1 (0) 0 1 1 0 0 1 1 0 0 1 1    -> 111011101110

%e 1 1 1 0 1 1 1 0 1 1 1 0 (1) 1 1 0 1 1 1 0 1 1 1 0 1  -> 1110111011101

%e (End)

%t f[n_] := Block[{w = {}}, Do[AppendTo[w, Boole[Mod[k, 4] != 3]], {k, 0, n}]; FromDigits@ w]; Table[f@ n, {n, 0, 19}] (* _Michael De Vlieger_, Dec 09 2015 *)

%o (

%Y Cf. A071037, A265381.

%K nonn,easy

%O 0,2

%A _Robert Price_, Dec 07 2015

%E Removed an unjustified claim that _Colin Barker_'s conjectures are correct. Removed 2 programs based on conjectures. - _N. J. A. Sloane_, Jun 13 2022