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Trajectory of x^3+x+1 under the map (see A185544) defined in the Comments.
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%I #12 Mar 30 2012 16:52:01

%S 1011,10011,101111,1001001,10110110,1011011,10011011,101111011,

%T 1001001011,10110110011,100110111111,1011110100001,10010010010010,

%U 1001001001001,10110110110110,1011011011011,10011011011011,101111011011011,1001001011011011,10110110011011011,100110111111011011,1011110100001011011

%N Trajectory of x^3+x+1 under the map (see A185544) defined in the Comments.

%C Analogous to A185000 except start with x^3+x+1.

%C This trajectory is a rare example where it can be proved that the trajectory diverges.

%C 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.

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

%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>

%e The trajectory begins x^3+x+1, 1+x+x^4, x^5+x^3+x^2+x+1, x^6+x^3+1, x^7+x^4+x+x^5+x^2, x^6+x^3+1+x^4+x, 1+x+x^3+x^4+x^7, x+x^4+x^5+x^8+1+x^3+x^6, 1+x+x^3+x^6+x^9, x+x^4+x^7+x^10+1+x^5+x^8,

%e x^11+x^8+x^7+x^5+x^4+x^3+x^2+x+1, x^12+x^10+x^9+x^8+x^7+x^5+1, x+x^4+x^7+x^10+x^13,

%e x^12+x^9+x^6+x^3+1, ...

%Y Cf. A185000, A185544.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Feb 05 2011