A309154 Function of natural numbers satisfying the properties a(2*n) = 2*a(n) and a(2*n+1) = -3 + 2*a(3*n+2).

0, 1, 2, 23, 4, 13, 46, 1595, 8, 6377, 26, 799, 92, 101, 3190, 3283, 16, 401, 12754, 12775, 52, 61, 1598, 1643, 184, 51097, 202, 946891009738223808271, 6380, 6389, 6566, 118361376217277976035, 32, 204385, 802, 823, 25508, 25517, 25550, 6540635, 104, 473445504869111904137

Offset

0,3

Comments

This integer sequence exists if and only if the Collatz conjecture is true. The proof is relatively trivial.

This is -3 times the Q function from Rozier restricted to the natural numbers.

The only multiple of 3 in the sequence is 0.

Links

Richard N. Smith, Table of n, a(n) for n = 0..2000

Olivier Rozier, Parity sequences of the 3x+1 map on the 2-adic integers and Euclidean embedding, arXiv:1805.00133 [math.DS], 2018.

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

Formula

a(2*n) = 2*a(n); a(2*n+1) = -3 + 2*a(3*n+2).

a(n) = -3*(n mod 2) + 2*a(A014682(n)) where A014682 is the Collatz function.

Example

For n = 0, the equation a(0) = 2*a(0) implies a(0) = 0.
For n = 1, the equation becomes a(1) = -3 + 2*a(2) = -3 + 4*a(1), so a(1) = 1.
For n = 3, a bit more calculating gives a(3) = -3 + 2*a(5) = -9 + 4*a(8) = -9 + 32*a(1) = 23.

Maple

a:= proc(n) option remember; `if`(n<2, n,
      `if`(irem(n, 2, 'r')=0, 2*a(r), 2*a(n+r+1)-3))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 14 2019

Mathematica

a[n_] := a[n] = If[n < 2, n, If[EvenQ[n], 2 a[n/2], 2 a[(3n + 1)/2] - 3]];
a /@ Range[0, 50] (* Jean-François Alcover, Sep 28 2019 *)

Prog

(Python)
def a(x):
    if x <= 1: return x
    elif x%2: return -3 + 2 * a((3*x + 1)//2)
    else: return 2*a(x//2)
(PARI) a(n)=if(n<=1, n, if(n%2, -3 + 2*a((3*n+1)/2), 2*a(n/2))) \\ Richard N. Smith, Jul 16 2019

Crossrefs

Cf. A014682.

Sequence in context: A245628 A162711 A120713 * A167920 A237579 A104644

Adjacent sequences: A309151 A309152 A309153 * A309155 A309156 A309157

Keyword

easy,nonn

Author

Jan Met den Ancxt, Jul 14 2019

Status

approved