A153727 Period 3: repeat [1, 4, 2] ; Trajectory of 3x+1 sequence starting at 1.

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, 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

Offset

0,2

Comments

From Klaus Brockhaus, May 23 2010: (Start)

Continued fraction expansion of (7+sqrt(229))/18.

Decimal expansion of 142/999. (End)

a(A008585(n)) = 1; a(A016777(n)) = 4; a(A016789(n)) = 2. - Reinhard Zumkeller, Oct 08 2011

References

C. A. Pickover, The Math Book, 2009; Collatz Conjecture, pp 374-375.

Links

Table of n, a(n) for n=0..104.

Eric Weisstein's World of Mathematics, Collatz Problem

Wikipedia, Collatz conjecture

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

Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Formula

a(3n)=1, a(3n+1)=4, a(3n+2)=2.

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

a(n) = 4^n mod 7. - Zerinvary Lajos, Nov 25 2009

a(n) = A130196(n) * A131534(n). - Reinhard Zumkeller, Nov 12 2009

a(n) = 2^(-n mod 3) = 2^A080425(n). - Wesley Ivan Hurt, Jun 20 2014

a(n) = sqrt(4^(5*n) mod 21). - Gary Detlefs, Jul 07 2014

From Wesley Ivan Hurt, Jun 30 2016: (Start)

a(n) = a(n-3) for n>2.

a(n) = (7 - 4*cos(2*n*Pi/3) + 2*sqrt(3)*sin(2*n*Pi/3))/3. (End)

Maple

A153727:=n->2^(-n mod 3); seq(A153727(n), n=0..50); # Wesley Ivan Hurt, Jun 20 2014

Mathematica

a[n_] := {1, 4, 2}[[Mod[n, 3] + 1]]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Jul 19 2013 *)
LinearRecurrence[{0, 0, 1}, {1, 4, 2}, 105] (* Ray Chandler, Aug 25 2015 *)

Prog

(Sage) [power_mod(2, -n, 7)for n in range(0, 105)] # Zerinvary Lajos, Jun 07 2009
(Sage) [power_mod(4, n, 7)for n in range(0, 105)] # Zerinvary Lajos, Nov 25 2009
(PARI) A153727(n)=[1, 4, 2][n%3+1] \\ M. F. Hasler, Feb 10 2011
(Haskell)
a153727 n = a153727_list !! n
a153727_list = iterate a006370 1 -- Reinhard Zumkeller, Oct 08 2011
(Magma) [2^(-n mod 3) : n in [0..50]]; // Wesley Ivan Hurt, Jun 20 2014

Crossrefs

Row 1 of A347270.

Cf. A178236 (decimal expansion of (7+sqrt(229))/18). Appears in A179133 (n>=1).

Cf. A006370, A080425, A130196, A131534.

Sequence in context: A082901 A136619 A132708 * A349245 A356631 A210937

Adjacent sequences: A153724 A153725 A153726 * A153728 A153729 A153730

Keyword

nonn,easy

Author

Philippe Deléham, Dec 30 2008

Status

approved