A309154 - OEIS

%I #32 Sep 28 2019 08:37:31

%S 0,1,2,23,4,13,46,1595,8,6377,26,799,92,101,3190,3283,16,401,12754,

%T 12775,52,61,1598,1643,184,51097,202,946891009738223808271,6380,6389,

%U 6566,118361376217277976035,32,204385,802,823,25508,25517,25550,6540635,104,473445504869111904137

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

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

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

%C The only multiple of 3 in the sequence is 0.

%H Richard N. Smith, <a href="/A309154/b309154.txt">Table of n, a(n) for n = 0..2000</a>

%H Olivier Rozier, <a href="https://arxiv.org/pdf/1805.00133.pdf">Parity sequences of the 3x+1 map on the 2-adic integers and Euclidean embedding</a>, arXiv:1805.00133 [math.DS], 2018.

%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>

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

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

%e For n = 0, the equation a(0) = 2*a(0) implies a(0) = 0.

%e For n = 1, the equation becomes a(1) = -3 + 2*a(2) = -3 + 4*a(1), so a(1) = 1.

%e For n = 3, a bit more calculating gives a(3) = -3 + 2*a(5) = -9 + 4*a(8) = -9 + 32*a(1) = 23.

%p a:= proc(n) option remember; `if`(n<2, n,

%p       `if`(irem(n, 2, 'r')=0, 2*a(r), 2*a(n+r+1)-3))

%p     end:

%p seq(a(n), n=0..40);  # _Alois P. Heinz_, Jul 14 2019

%t a[n_] := a[n] = If[n < 2, n, If[EvenQ[n], 2 a[n/2], 2 a[(3n + 1)/2] - 3]];

%t a /@ Range[0, 50] (* _Jean-François Alcover_, Sep 28 2019 *)

%o (Python)

%o def a(x):

%o     if x <= 1: return x

%o     elif x%2: return -3 + 2 * a((3*x + 1)//2)

%o     else: return 2*a(x//2)

%o (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

%Y Cf. A014682.

%K easy,nonn

%O 0,3

%A _Jan Met den Ancxt_, Jul 14 2019