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A185000 Trajectory of x+1 under the map (see A185544) defined in the Comments. 2
11, 111, 1101, 11100, 1110, 111, 1101, 11100, 1110, 111, 1101, 11100, 1110, 111, 1101, 11100, 1110, 111, 1101, 11100, 1110, 111, 1101, 11100, 1110, 111, 1101, 11100, 1110, 111, 1101, 11100, 1110, 111, 1101, 11100, 1110, 111, 1101, 11100, 1110, 111, 1101, 11100, 1110, 111, 1101, 11100, 1110, 111 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

We work in the ring GF(2)[x]. The map is f->f/x if f(0)=0, otherwise f->((x^2+1)f+1)/x. We represent polynomials by their vector of coefficients, high powers first. See A185544.

REFERENCES

J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010; see page 99.

LINKS

Table of n, a(n) for n=1..50.

Index entries for sequences related to 3x+1 (or Collatz) problem

EXAMPLE

The trajectory is x^2+x+1, x^3+x^2+1, x^4+x^3+x^2, x^3+x^2+x, x^2+x+1, x^3+x^2+1, x^4+x^3+x^2, x^3+x^2+x, x^2+x+1, x^3+x^2+1, ..., with period 4.

MAPLE

# Extract coefficient vector polynomial (decreasing powers):

coeflistD:=proc(f) local d, i, t1, t2, t3, t4;

if f=0 then RETURN([0]); else

d:=degree(f);

t1:=subs(x=1/x, f);

t2:=sort(expand(x^d*t1));

t3:=seriestolist(series(t2, x, d+2));

t4:=nops(t3);

if t4<d+1 then for i from t4+1 to d+1 do t3:=[op(t3), 0]; od: fi;

RETURN(t3);

fi;

end;

# Define map f:

f:=a->if subs(x=0, a) = 0 then expand(simplify(a/x)) mod 2;

else t1:=((x^2+1)*a+1)/x;  expand(t1) mod 2; fi;

# Get trajectory (as both polynomials and coefficient vectors):

T:=proc(n, M) global f, coeflistD; local t1, i, s1; t1:=[n];

for i from 1 to M-1 do t1:=[op(t1), f(t1[nops(t1)])]; od: lprint(t1);

s1:=[]; for i from 1 to M do s1:=[op(s1), coeflistD(t1[i])]; od: lprint(s1);

end;

T(x+1, 12);

CROSSREFS

Cf. A185544.

Sequence in context: A284024 A283175 A284274 * A283585 A283703 A284399

Adjacent sequences:  A184997 A184998 A184999 * A185001 A185002 A185003

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 05 2011

STATUS

approved

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Last modified November 17 19:40 EST 2017. Contains 294834 sequences.