A328010 The 5x + 1 sequence beginning at 17.

17, 86, 43, 216, 108, 54, 27, 136, 68, 34, 17, 86, 43, 216, 108, 54, 27, 136, 68, 34, 17, 86, 43, 216, 108, 54, 27, 136, 68, 34, 17, 86, 43, 216, 108, 54, 27, 136, 68, 34, 17, 86, 43, 216, 108, 54, 27, 136, 68, 34, 17, 86, 43, 216, 108, 54, 27, 136, 68, 34, 17

Offset: 0

Author

Antoine Beaulieu, Oct 01 2019

Keywords

• nonn
• easy

Comments

The 5x+1 problem is similar to the 3x+1 or Collatz problem. For some starting values it is known that the 5x+1 trajectory will tend to infinity or enter a periodic orbit.

Alex V. Kontorovich & Jeffrey C. Lagarias conjectured that there are very few periodic orbits. One of them is shown here.

The two other known periodic orbits are given in the crossrefs.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Alex V. Kontorovich & Jeffrey C. Lagarias, Stochastic Models for the 3x+1 and 5x+1 Problems arXiv:0910.1944 [math.NT], 2009.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).

Crossrefs

Cf. A259207, A328011.

Programs

PARI

Vec((17 + 86*x + 43*x^2 + 216*x^3 + 108*x^4 + 54*x^5 + 27*x^6 + 136*x^7 + 68*x^8 + 34*x^9) / ((1 - x)*(1 + x)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)) + O(x^60)) \\ Colin Barker, Oct 05 2019

Formula

a(n+1) = 5*a(n) + 1 if a(n) is odd, a(n+1) = a(n)/2 otherwise.

From Colin Barker, Oct 04 2019: (Start)

G.f.: (17 + 86*x + 43*x^2 + 216*x^3 + 108*x^4 + 54*x^5 + 27*x^6 + 136*x^7 + 68*x^8 + 34*x^9) / ((1 - x)*(1 + x)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)).

a(n) = a(n-10) for n>9.

(End)