login
A153727
Period 3: repeat [1, 4, 2] ; Trajectory of 3x+1 sequence starting at 1.
20
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.
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).
Sequence in context: A082901 A136619 A132708 * A349245 A356631 A210937
KEYWORD
nonn,easy
AUTHOR
Philippe Deléham, Dec 30 2008
STATUS
approved