A185881 Trajectory of x^3+x+1 under the map (see A185544) defined in the Comments.

1011, 10011, 101111, 1001001, 10110110, 1011011, 10011011, 101111011, 1001001011, 10110110011, 100110111111, 1011110100001, 10010010010010, 1001001001001, 10110110110110, 1011011011011, 10011011011011, 101111011011011, 1001001011011011, 10110110011011011, 100110111111011011, 1011110100001011011

Offset

1,1

Comments

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

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

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..22.

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

Example

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

Crossrefs

Cf. A185000, A185544.

Sequence in context: A345906 A035125 A183849 * A115769 A267613 A178396

Adjacent sequences: A185878 A185879 A185880 * A185882 A185883 A185884

Keyword

nonn

Author

N. J. A. Sloane, Feb 05 2011

Status

approved