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A266613 Decimal representation of the middle column of the "Rule 41" elementary cellular automaton starting with a single ON (black) cell.

%I A266613 #63 Feb 16 2025 08:33:28
%S A266613 1,2,5,10,20,41,82,165,330,661,1322,2645,5290,10581,21162,42325,84650,
%T A266613 169301,338602,677205,1354410,2708821,5417642,10835285,21670570,
%U A266613 43341141,86682282,173364565,346729130,693458261,1386916522,2773833045,5547666090,11095332181
%N A266613 Decimal representation of the middle column of the "Rule 41" elementary cellular automaton starting with a single ON (black) cell.
%H A266613 Robert Price, <a href="/A266613/b266613.txt">Table of n, a(n) for n = 0..1000</a>
%H A266613 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A266613 Stephen Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>, Wolfram Media, 2002; p. 55.
%H A266613 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A266613 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A266613 A266612(n) = A007088(a(n)).
%F A266613 Conjectures from _Colin Barker_, Jan 02 2016 and Apr 16 2019: (Start)
%F A266613 a(n) = (31*2^n-4*((-1)^n+3))/24 for n>2.
%F A266613 a(n) = 2*a(n-1)+a(n-2)-2*a(n-3) for n>5. - [corrected by _Karl V. Keller, Jr._, Oct 07 2021]
%F A266613 G.f.: (1-x^4+x^5) / ((1-x)*(1+x)*(1-2*x)). (End)
%F A266613 Conjecture: a(n) = A000975(n) + 20*2^(n-5), for n>2. - _Andres Cicuttin_, Mar 31 2016
%p A266613 # Rule 41: value in generation r and column c, where c=0 is the central one
%p A266613 r41 := proc(r::integer,c::integer)
%p A266613     option remember;
%p A266613     local up ;
%p A266613     if r = 0 then
%p A266613         if c = 0 then
%p A266613             1;
%p A266613         else
%p A266613             0;
%p A266613         end if;
%p A266613     else
%p A266613         # previous 3 bits
%p A266613         [procname(r-1,c+1),procname(r-1,c),procname(r-1,c-1)] ;
%p A266613         up := op(3,%)+2*op(2,%)+4*op(1,%) ;
%p A266613         # rule 41 = 00101001_2: {5,3,0}->1, all others ->0
%p A266613         if up in {5,3,0} then
%p A266613             1;
%p A266613         else
%p A266613             0 ;
%p A266613         end if;
%p A266613     end if;
%p A266613 end proc:
%p A266613 A266613 := proc(n)
%p A266613     b := [seq(r41(r,0),r=0..n)] ;
%p A266613     add(op(-i,b)*2^(i-1),i=1..nops(b)) ;
%p A266613 end proc:
%p A266613 smax := 20 ;
%p A266613 L := [seq(A266613(n),n=0..smax)] ; # _R. J. Mathar_, Apr 12 2019
%t A266613 rule=41; 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 *) mc=Table[catri[[k]][[k]],{k,1,rows}]; (* Keep only middle cell from each row *) Table[FromDigits[Take[mc,k],2],{k,1,rows}] (* Binary Representation of Middle Column *)
%Y A266613 Cf. A000975, A266608, A266611, A266612, A081253.
%K A266613 nonn,easy
%O A266613 0,2
%A A266613 _Robert Price_, Jan 01 2016
%E A266613 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