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

Author

Jeff Burch

Keywords

• nonn
• easy

Examples

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

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).

Crossrefs

Cf. A070165.

Row 9 of A347270.

Programs

Mathematica

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

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

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

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)