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

Author

Alonso del Arte, Jun 20 2015

Keywords

• nonn
• easy

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.

Examples

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

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. A033478, A245671.

Programs

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

Mathematica

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

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

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)