A033479 3x+1 sequence beginning at 9.

9, 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, 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

0,1

Links

Colin Barker, Table of n, a(n) for n = 0..1000

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

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

Formula

From Colin Barker, Oct 04 2019: (Start)

G.f.: (9 + 28*x + 14*x^2 - 2*x^3 - 6*x^4 - 3*x^5 + 27*x^6 - 5*x^7 + 41*x^8 - 8*x^9 - 4*x^10 - 12*x^11 - 6*x^12 - 3*x^13 - 35*x^14 - 4*x^15 - 2*x^16 - x^17 - 14*x^18 - 7*x^19) / ((1 - x)*(1 + x + x^2)).

a(n) = a(n-3) for n>19.

(End)

Example

9 is odd, so the next term is 3*9 + 1 = 28.
28 is even, so the next term is 28/2 = 14.

Mathematica

NestList[If[EvenQ[#], #/2, 3# + 1] &, 9, 100] (* Harvey P. Dale, Dec 16 2012 *)

Prog

(Scala) def collatz(n: Int): Int = (n % 2) match {
  case 0 => n / 2
  case 1 => 3 * n + 1
}
import scala.collection.mutable.ListBuffer
val start = 9; var curr = start; var trajectory = new ListBuffer[Int]()
for (_ <- 1 to 100) {
  trajectory += curr; curr = collatz(curr)
}
trajectory // Alonso del Arte, Jun 02 2019
(PARI) Vec((9 + 28*x + 14*x^2 - 2*x^3 - 6*x^4 - 3*x^5 + 27*x^6 - 5*x^7 + 41*x^8 - 8*x^9 - 4*x^10 - 12*x^11 - 6*x^12 - 3*x^13 - 35*x^14 - 4*x^15 - 2*x^16 - x^17 - 14*x^18 - 7*x^19) / ((1 - x)*(1 + x + x^2)) + O(x^80)) \\ Colin Barker, Oct 04 2019
(Python)
from itertools import accumulate
def f(x, _): return x//2 if x%2 == 0 else 3*x+1
print(list(accumulate([9]*92, f))) # Michael S. Branicky, Sep 28 2021

Crossrefs

Cf. A070165.

Row 9 of A347270.

Sequence in context: A351266 A369721 A369759 * A254139 A034116 A031454

Adjacent sequences: A033476 A033477 A033478 * A033480 A033481 A033482

Keyword

nonn,easy

Author

Jeff Burch

Status

approved