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

1, 2, 5, 10, 20, 41, 82, 165, 330, 661, 1322, 2645, 5290, 10581, 21162, 42325, 84650, 169301, 338602, 677205, 1354410, 2708821, 5417642, 10835285, 21670570, 43341141, 86682282, 173364565, 346729130, 693458261, 1386916522, 2773833045, 5547666090, 11095332181

Offset: 0

Author

Robert Price, Jan 01 2016

Keywords

• nonn
• easy

Links

• Robert Price, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
• Index entries for sequences related to cellular automata
• Index to Elementary Cellular Automata

Crossrefs

Cf. A000975, A266608, A266611, A266612, A081253.

Programs

Maple

# Rule 41: value in generation r and column c, where c=0 is the central one
r41 := proc(r::integer,c::integer)
    option remember;
    local up ;
    if r = 0 then
        if c = 0 then
            1;
        else
            0;
        end if;
    else
        # previous 3 bits
        [procname(r-1,c+1),procname(r-1,c),procname(r-1,c-1)] ;
        up := op(3,%)+2*op(2,%)+4*op(1,%) ;
        # rule 41 = 00101001_2: {5,3,0}->1, all others ->0
        if up in {5,3,0} then
            1;
        else
            0 ;
        end if;
    end if;
end proc:
A266613 := proc(n)
    b := [seq(r41(r,0),r=0..n)] ;
    add(op(-i,b)*2^(i-1),i=1..nops(b)) ;
end proc:
smax := 20 ;
L := [seq(A266613(n),n=0..smax)] ; # R. J. Mathar, Apr 12 2019

Mathematica

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 *)

Formula

A266612(n) = A007088(a(n)).

Conjectures from Colin Barker, Jan 02 2016 and Apr 16 2019: (Start)

a(n) = (31*2^n-4*((-1)^n+3))/24 for n>2.

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]

G.f.: (1-x^4+x^5) / ((1-x)*(1+x)*(1-2*x)). (End)

Conjecture: a(n) = A000975(n) + 20*2^(n-5), for n>2. - Andres Cicuttin, Mar 31 2016

Extensions

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