A075884 Image of n at the second step of the 3x+1 algorithm.

2, 4, 5, 1, 8, 10, 11, 2, 14, 16, 17, 3, 20, 22, 23, 4, 26, 28, 29, 5, 32, 34, 35, 6, 38, 40, 41, 7, 44, 46, 47, 8, 50, 52, 53, 9, 56, 58, 59, 10, 62, 64, 65, 11, 68, 70, 71, 12, 74, 76, 77, 13, 80, 82, 83, 14, 86, 88, 89, 15, 92, 94, 95, 16, 98, 100, 101, 17, 104, 106, 107

Offset: 1

Author

Bruce Corrigan (scentman(AT)myfamily.com), Oct 16 2002

Keywords

• easy
• nonn

Comments

Also known as the Collatz Problem, Syracuse Algorithm or Hailstone Problem. Let syr(m,n) be the image of n at the m-th step. for m=2, k>=0 we get: syr(2,4k)=k, syr(2,4k+1)=6k+2, syr(2,4k+2)=6k+4, syr(2,4k+3)=6k+5.

Examples

			1->4->2, 2->1->4, 3->10->5, 4->2->1, ...
		

References

• David Wells, Penguin Dictionary of Curious and Interesting Numbers

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Collatz Problem
• Index entries for sequences related to 3x+1 (or Collatz) problem
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Crossrefs

Cf. A006370 (the sequence at step 1), A076536 (at step 3).

Column k=2 of A347270.

Programs

Magma

m:=80; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(x^6+2*x^5+4*x^4+x^3+5*x^2+4*x+2)/(1-x^4)^2));

Mathematica

Table[Nest[If[EvenQ[#],#/2,3#+1]&,n,2],{n,80}] (* Harvey P. Dale, Nov 15 2012 *)

PARI

x='x+O('x^80); Vec(x*(x^6+2*x^5+4*x^4+x^3+5*x^2+4*x+2)/(1-x^4)^2) \\ G. C. Greubel, Oct 16 2018

Formula

G.f.: x*(x^6 +2*x^5 +4*x^4 +x^3 +5*x^2 +4*x +2)/(1-x^4)^2.

a(n) = (6*n +(55*n+4)*m -6*(5*n-2)*m^2 +(5*n-4)*m^3)/24, m=(n mod 4). - Zak Seidov, Sep 14 2006

From Federico Provvedi, Oct 17 2021: (Start)

Dirichlet g.f.: ((3*4^s - 10)*zeta(s-1) + (4^s + 2^s - 2)*zeta(s))/2^(2s+1).

a(n) = (n*(19-5*i^(2*n)) - (5*n+4)*(i^n + (-i)^n) + 8)/16, where i*i = -1. (End)

a(n) = 2*a(n-4) - a(n-8). - Wesley Ivan Hurt, Apr 16 2023