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A267515 Decimal representation of the middle column of the "Rule 137" elementary cellular automaton starting with a single ON (black) cell. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A261681 A030525 A116385 * A215411 A243988 A279811

Adjacent sequences:  A267512 A267513 A267514 * A267516 A267517 A267518

KEYWORD

nonn,easy

AUTHOR

Robert Price, Jan 16 2016

STATUS

approved

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Last modified August 20 06:25 EDT 2019. Contains 326139 sequences. (Running on oeis4.)