A122563 Start at 2n+1, iterate the map x -> A337349(x); sequence gives the number of iterations to resulting cycle or -1 if the trajectory never cycles.

0, 2, 1, 1, 2, 3, 2, 1, 0, 0, 1, 3, 2, 1, 1, 6, 3, 5, 2, 4, 4, 0, 3, 3, 8, 8, 2, 7, 1, 4, 0, 3, 6, 3, 1, 2, 5, 10, 1, 4, 10, 7, 1, 9, 3, 9, 3, 8, 0, 8, 2, 2, 5, 7, 0, 7, 7, 7, 1, 4, 1, 2, 6, 6, 6, 9, 3, 1, 2, 5, 5, 5, 5, 8, 2, 2, 1, 10, 4, 16, 4, 4, 4, 4, 9, 6, 1, 9, 3, 15, 3, 3, 3, 6, 3, 3, 2, 8, 8, 2, 8, 14

Offset

0,2

Comments

Iteration: multiply by 3 and add 1 and divide out any power of 2; then multiply by 3 and subtract 1 and divide out any power of 2.

When a(x) is iterated, what are the limit cycles? Are there any besides {1} and {17 -> 19 -> 43 -> 97 -> 109 -> 61}?

Links

Ray Chandler, Table of n, a(n) for n = 0..10000

Example

The iteration for n=13 is 27->61->17->19->43->97->109->61->... and a(13)=1 step was needed to enter the cycle (at 61).
The iteration for n=30 is 61-> 17->19->43->97->109->61->> and the cycle was already entered at the start, so a(30)=0.

Maple

A122563 := proc(n)
    local cyc, itr, x ;
    cyc := [] ;
    x := 2*n+1 ;
    while true do
        cyc := [op(cyc), x] ;
        x := A337349(x) ;
        if x in cyc then
            break ;
        end if;
    end do:
    member(x, cyc, 'itr') ;
    itr -1 ;
end proc:
seq(A122563(n), n=0..101) ; # R. J. Mathar, Aug 26 2020

Mathematica

nextx[x_Integer] := Block[{a = x}, a = 3 a + 1; While[EvenQ@a, a /= 2]; a = 3 a - 1; While[EvenQ@a, a /= 2]; a]; f[n_] := Length@NestWhileList[nextx, n, FreeQ[{1, 17, 19, 43, 97, 109, 61}, #] &] - 1; Table[f[2 n + 1], {n, 0, 101}] (* original program from author corrected as suggested by William P. Orrick, Ray Chandler, Aug 28 2020 *)

Crossrefs

Cf. A102421, A102423, A337349.

Sequence in context: A063740 A072782 A337014 * A204030 A234503 A333267

Adjacent sequences: A122560 A122561 A122562 * A122564 A122565 A122566

Keyword

nonn,look

Author

Robert G. Wilson v, based on email from Dan Asimov (dasimov(AT)earthlink.net), Sep 20 2006

Extensions

a(13), a(30),... corrected. - R. J. Mathar, Aug 26 2020

Status

approved