A259207 5x + 1 sequence beginning at 5.

5, 26, 13, 66, 33, 166, 83, 416, 208, 104, 52, 26, 13, 66, 33, 166, 83, 416, 208, 104, 52, 26, 13, 66, 33, 166, 83, 416, 208, 104, 52, 26, 13, 66, 33, 166, 83, 416, 208, 104, 52, 26, 13, 66, 33, 166, 83, 416, 208, 104, 52, 26, 13, 66, 33, 166, 83, 416, 208, 104, 52, 26, 13, 66, 33

Offset

0,1

Comments

It's still not known whether every 3x + 1 sequence reaches 1. But for the 5x + 1 variant, the answer is clearly no, as this sequence demonstrates: 26 is first encountered as 5 * 5 + 1, but every time afterwards as half 52.

However, there are still unanswered questions about the 5x + 1 problem. Kontorovich and Lagarias (2009) say that it is conjectured that there are very few periodic orbits, one of which is the one exhibited by this sequence.

Some authors call "5x+1" the map that would divide x by 3 if it's an odd multiple of 3, see A057688 for the trajectory of 5 under that map. - M. F. Hasler, Feb 11 2026

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(0) = 5; a(n) = 5*a(n - 1) + 1 if a(n - 1) is odd, a(n) = a(n - 1)/2 otherwise.

From Colin Barker, Oct 04 2019: (Start)

G.f.: (5 + 26*x + 13*x^2 + 66*x^3 + 33*x^4 + 166*x^5 + 83*x^6 + 416*x^7 + 208*x^8 + 104*x^9 + 47*x^10) / ((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>10.

(End)

Example

5 is odd, so it's followed by 5 * 5 + 1 = 26.
26 is even, so it's followed by 26/2 = 13.

Mathematica

NestList[If[EvenQ[#], #/2, 5# + 1] &, 5, 100]

Prog

(Magma) [n eq 1 select 5 else IsOdd(Self(n-1)) select 5*Self(n-1)+1 else Self(n-1) div 2: n in [1..100]]; // Vincenzo Librandi, Jun 21 2015
(PARI) Vec((5 + 26*x + 13*x^2 + 66*x^3 + 33*x^4 + 166*x^5 + 83*x^6 + 416*x^7 + 208*x^8 + 104*x^9 + 47*x^10) / ((1 - x)*(1 + x)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)) + O(x^50)) \\ Colin Barker, Oct 04 2019
(PARI)
{A259207(n, x=5)=n&& for(i=1, (n-1)%10+1, x=if(x%2, x*5+1, x\2)); x} \\ optional 2nd arg to use a different starting value; e.g. x=11 gives A259193.
A259207_first(n)=vector(n, i, n=if(i<2, 5, n%2, n*5+1, n\2)) \\ M. F. Hasler, Feb 11 2026

Crossrefs

Cf. A033478, A245671.

Cf. A057688 (trajectory of 5 under a different 5x+1 map).

Sequence in context: A060063 A106295 A057688 * A300005 A048269 A073069

Adjacent sequences: A259204 A259205 A259206 * A259208 A259209 A259210

Keyword

nonn,easy

Author

Alonso del Arte, Jun 20 2015

Status

approved