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

0,2

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

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

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 *)

Crossrefs

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

Sequence in context: A024810 A049938 A002460 * A266462 A293319 A267589

Adjacent sequences: A266610 A266611 A266612 * A266614 A266615 A266616

Keyword

nonn,easy

Author

Robert Price, Jan 01 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

Status

approved