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

0,1

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).

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)

Mathematica

NestList[If[OddQ[#], 5*# + 1, #/2] &, 17, 99] (* Paolo Xausa, Jul 08 2026 *)

Prog

(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

Crossrefs

Cf. A259207, A328011.

Sequence in context: A288420 A156157 A146389 * A041554 A080770 A118863

Adjacent sequences: A328007 A328008 A328009 * A328011 A328012 A328013

Keyword

nonn,easy,changed

Author

Antoine Beaulieu, Oct 01 2019

Status

approved