A008876 3x+1 sequence starting at 81.

81, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• R. K. Guy, Unsolved Problems in Number Theory, E16.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Darrell Cox, The 3n + 1 Problem: A Probabilistic Approach, Journal of Integer Sequences, Vol. 15 (2012), #12.5.2.
• Index entries for sequences related to 3x+1 (or Collatz) problem
• Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Crossrefs

Cf. A006370.

Row 81 of A347270.

Programs

Haskell

a008876 n = a008876_list !! n
a008876_list = 81 : iterate a006370 81
-- Reinhard Zumkeller, Aug 30 2012

Magma

[n eq 1 select 81 else IsOdd(Self(n-1)) select 3*Self(n-1)+1 else Self(n-1) div 2: n in [1..70]]; // Vincenzo Librandi, Jul 29 2014

Maple

f := proc(n) option remember; if n = 0 then 81; elif f(n-1) mod 2 = 0 then f(n-1)/2 else 3*f(n-1)+1; fi; end;

Mathematica

NestList[If[EvenQ[#], #/2, 3# + 1]&, 81, 100] (* Vincenzo Librandi, Jul 29 2014 *)

PARI

Vec((81 + 244*x + 122*x^2 - 20*x^3 - 60*x^4 - 30*x^5 - 15*x^6 - 161*x^7 - 22*x^8 - 11*x^9 + 83*x^10 - 17*x^11 + 125*x^12 - 26*x^13 - 13*x^14 - 140*x^15 - 70*x^16 - 35*x^17 - 4*x^18 - 2*x^19 - x^20 - 14*x^21 - 7*x^22) / ((1 - x)*(1 + x + x^2)) + O(x^70)) \\ Colin Barker, Apr 27 2020

Scala

def collatz(n: Int): Int = n % 2 match { case 0 => n / 2; case _ => 3 * n + 1 }
def collatzSeq(n: Int): LazyList[Int] = LazyList.iterate(n)(collatz)
collatzSeq(81).take(100).toList // Alonso del Arte, Apr 24 2020

Formula

From Colin Barker, Apr 27 2020: (Start)

G.f.: (81 + 244*x + 122*x^2 - 20*x^3 - 60*x^4 - 30*x^5 - 15*x^6 - 161*x^7 - 22*x^8 - 11*x^9 + 83*x^10 - 17*x^11 + 125*x^12 - 26*x^13 - 13*x^14 - 140*x^15 - 70*x^16 - 35*x^17 - 4*x^18 - 2*x^19 - x^20 - 14*x^21 - 7*x^22) / ((1 - x)*(1 + x + x^2)).

a(n) = a(n-3) for n>22.

(End)