A008879 3x+1 sequence starting at 87.

87, 262, 131, 394, 197, 592, 296, 148, 74, 37, 112, 56, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 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

Offset

0,1

Comments

Also known as the Collatz sequence. - Harvey P. Dale, Feb 03 2026

References

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for sequences related to 3x+1 (or Collatz) problem

Index entries for periodic sequences.

Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Formula

G.f.: (87 + 262*x + 131*x^2 + 307*x^3 - 65*x^4 + 461*x^5 - 98*x^6 - 49*x^7 - 518*x^8 - 259*x^9 - 36*x^10 - 18*x^11 - 9*x^12 - 98*x^13 - 49*x^14 - 6*x^15 - 3*x^16 + 27*x^17 - 5*x^18 + 41*x^19 - 8*x^20 - 4*x^21 - 12*x^22 - 6*x^23 - 3*x^24 - 35*x^25 - 4*x^26 - 2*x^27 - x^28 - 14*x^29 - 7*x^30)/(1 - x^3). - Charles R Greathouse IV, May 17 2026

Maple

f := proc(n) option remember; if n = 0 then 87; 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]&, 87, 100] (* Vincenzo Librandi, Jul 29 2014 *)

Prog

(Magma) [n eq 1 select 87 else IsOdd(Self(n-1)) select 3*Self(n-1)+1 else Self(n-1) div 2: n in [1..80]]; // Vincenzo Librandi, Jul 29 2014
(PARI) a(n)=if(n>27, [1, 4, 2][n%3+1], [87, 262, 131, 394, 197, 592, 296, 148, 74, 37, 112, 56, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8][n+1]) \\ Charles R Greathouse IV, May 17 2026
(PARI) a(n)=if(n>27, ([0, 1, 0; 0, 0, 1; 1, 0, 0]^(n-28)*[4; 2; 1])[1, 1], [87, 262, 131, 394, 197, 592, 296, 148, 74, 37, 112, 56, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8][n+1]) \\ Charles R Greathouse IV, May 26 2026

Crossrefs

Cf. similar sequences listed in A245671.

Row 87 of A347270.

Sequence in context: A304509 A020314 A008899 * A186063 A186055 A243580

Adjacent sequences: A008876 A008877 A008878 * A008880 A008881 A008882

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved