A266753 - OEIS

%I #48 Jun 13 2022 08:56:08

%S 1,4,18,74,298,1194,4778,19114,76458,305834,1223338,4893354,19573418,

%T 78293674,313174698,1252698794,5010795178,20043180714,80172722858,

%U 320690891434,1282763565738,5131054262954,20524217051818,82096868207274,328387472829098

%N Decimal representation of the n-th iteration of the "Rule 163" 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="/A266753/b266753.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://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 Conjectures from _Colin Barker_, Jan 20 2016 and Apr 20 2019: (Start)

%F a(n) = 5*a(n-1)-4*a(n-2) for n>2.

%F G.f.: (1-x+2*x^2) / ((1-x)*(1-4*x)).

%F (End)

%F Empirical a(n) = (7*4^n - 4)/6 for n>1. - _Colin Barker_, Nov 25 2016 and Apr 20 2019

%F a(n) = 4*a(n-1) + 2, n>1, conjectured. - _Yosu Yurramendi_, Jan 22 2017

%F a(n) = 2*A020988(n) - A020988(n-1) = A020988(n) + 2^(2n-1) for n > 0, conjectured. - _Yosu Yurramendi_, Jan 24 2017 [n range correction - _Karl V. Keller, Jr._, May 07 2022]

%F a(n) = A072197(n-1) + A002450(n), n > 0, conjectured. - _Yosu Yurramendi_, Mar 03 2017

%t rule=163; 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]],2],{k,1,rows}]   (* Decimal Representation of Rows *)

%Y Cf. A263919, A266752.

%K nonn,easy

%O 0,2

%A _Robert Price_, Jan 17 2016

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