A267515 Decimal representation of the middle column of the "Rule 137" elementary cellular automaton starting with a single ON (black) cell.

1, 2, 5, 10, 21, 42, 84, 169, 338, 677, 1354, 2709, 5418, 10836, 21673, 43346, 86693, 173386, 346773, 693546, 1387092, 2774185, 5548370, 11096741, 22193482, 44386965, 88773930, 177547860, 355095721, 710191442, 1420382885, 2840765770, 5681531541, 11363063082

Offset

0,2

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

[Incorrect] conjectures from Colin Barker, Jan 16 2016: (Start)

a(n) = 2*a(n-1)+a(n-7)-2*a(n-8) for n>7. [Wrong]

G.f.: (1-x+x^2)*(1+x+x^2) / ((1-x)*(1-2*x)*(1+x+x^2+x^3+x^4+x^5+x^6)). [Wrong]

(End)

The linear recurrence and g.f. are invalid and start to generate erroneous values at a(62). - R. J. Mathar, Apr 12 2019

Maple

# Rule 137: value in generation r and column c, where c=0 is the central one
r137 := 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 137 = 10001001_2: 7->1, {6, 5, 4}->0, 3->1, {2, 1}->0, 0->1
        if up in {7, 3, 0} then
            1;
        else
            0 ;
        end if;
    end if;
end proc:
A267515 := proc(n)
    b := [seq(r137(r, 0), r=0..n)] ;
    add(2^(i-1)*op(-i, b), i=1..nops(b)) ;
end proc:
[seq(A267515(n), n=0..62)] ; # R. J. Mathar, Apr 12 2019

Mathematica

rule=137; 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. A267463, A267514.

Sequence in context: A333798 A030525 A116385 * A215411 A243988 A279811

Adjacent sequences: A267512 A267513 A267514 * A267516 A267517 A267518

Keyword

nonn,easy

Author

Robert Price, Jan 16 2016

Status

approved