A267678 - OEIS

%I #25 Jun 14 2022 08:18:58

%S 1,2,20,104,468,1960,8020,32424,130388,522920,2094420,8383144,

%T 33543508,134195880,536827220,2147396264,8589759828,34359388840,

%U 137438254420,549754415784,2199020459348,8796087429800,35184360904020,140737465985704,562949908682068

%N Decimal representation of the n-th iteration of the "Rule 197" 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="/A267678/b267678.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 19 2016: (Start)

%F a(n) = 6*a(n-1)-7*a(n-2)-6*a(n-3)+8*a(n-4) for n>5.

%F G.f.: (1-4*x+15*x^2+4*x^3-12*x^4-16*x^5) / ((1-x)*(1+x)*(1-2*x)*(1-4*x)).

%F (End)

%F Conjecture: a(n) = 2*(-3 + (-1)^n - 2^(2+n) + 3*4^n)/3 for n>1. - _Colin Barker_, Feb 14 2017

%F Conjecture: a(n) = floor((6*4^n - 8*2^n)/3) - 2^(n mod 2) for n > 1. - _Karl V. Keller, Jr._, Jun 01 2022

%t rule=197; 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. A267676, A267677.

%K nonn

%O 0,2

%A _Robert Price_, Jan 19 2016