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

Author

Robert Price, Jan 18 2016

Keywords

• nonn
• easy

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

Crossrefs

Cf. A267621, A283508, A083329.

Programs

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 *)

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

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