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 A259207 5x + 1 sequence beginning at 5. 9
 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 (list; graph; refs; listen; history; text; internal format)
 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. 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 CROSSREFS Cf. A033478, A245671. 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

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Last modified September 20 11:47 EDT 2021. Contains 347584 sequences. (Running on oeis4.)