A259207 - OEIS

%I #15 Sep 08 2022 08:46:13

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

%T 52,26,13,66,33,166,83,416,208,104,52,26,13,66,33,166,83,416,208,104,

%U 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

%N 5x + 1 sequence beginning at 5.

%C 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.

%C 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.

%H Colin Barker, <a href="/A259207/b259207.txt">Table of n, a(n) for n = 0..1000</a>

%H Alex V. Kontorovich & Jeffrey C. Lagarias, <a href="http://arxiv.org/abs/0910.1944">Stochastic Models for the 3x+1 and 5x+1 Problems</a> arXiv:0910.1944 [math.NT], 2009.

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,1).

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

%F From _Colin Barker_, Oct 04 2019: (Start)

%F 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)).

%F a(n) = a(n-10) for n>10.

%F (End)

%e 5 is odd, so it's followed by 5 * 5 + 1 = 26.

%e 26 is even, so it's followed by 26/2 = 13.

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

%o (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

%o (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

%Y Cf. A033478, A245671.

%K nonn,easy

%O 0,1

%A _Alonso del Arte_, Jun 20 2015