A267623 Binary representation of the middle column of the "Rule 187" elementary cellular automaton starting with a single ON (black) cell.

1, 10, 101, 1011, 10111, 101111, 1011111, 10111111, 101111111, 1011111111, 10111111111, 101111111111, 1011111111111, 10111111111111, 101111111111111, 1011111111111111, 10111111111111111, 101111111111111111, 1011111111111111111, 10111111111111111111

Offset

0,2

Comments

Also, The binary representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. See A283508.

References

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

Robert Price, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

S. Wolfram, A New Kind of Science

Index entries for sequences related to cellular automata

Index to Elementary Cellular Automata

Formula

Conjectures from Colin Barker, Jan 19 2016 and Apr 16 2019: (Start)

a(n) = 11*a(n-1)-10*a(n-2) for n>2.

G.f.: (1-x+x^2) / ((1-x)*(1-10*x)).

(End)

Empirical: a(n) = (91*10^n - 10) / 90 for n>0. - Colin Barker, Mar 10 2017

It also appears that a(n) = floor(91*10^n/90). - Karl V. Keller, Jr., May 28 2022

Maple

# Rule 187: value in generation r and column c, where c=0 is the central one
r187 := 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 187 = 10111011_2: {6, 2}->0, all others ->1
        if up in {6, 2} then
            0;
        else
            1 ;
        end if;
    end if;
end proc:
A267623 := proc(n)
    b := [seq(r187(r, 0), r=0..n)] ;
    add(op(-i, b)*2^(i-1), i=1..nops(b)) ;
    A007088(%) ;
end proc:
smax := 30 ;
L := [seq(A267623(n), n=0..smax)] ; # R. J. Mathar, Apr 12 2019

Mathematica

rule=187; 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]], {k, 1, rows}]  (* Binary Representation of Middle Column *)

Crossrefs

Cf. A267621, A283508, A083329.

Sequence in context: A033128 A094945 A190480 * A283508 A261200 A175541

Adjacent sequences: A267620 A267621 A267622 * A267624 A267625 A267626

Keyword

nonn,easy

Author

Robert Price, Jan 18 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