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

%I A153727 #61 Jul 20 2025 15:45:29
%S A153727 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,
%T A153727 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,
%U A153727 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
%N A153727 Period 3: repeat [1, 4, 2] ; Trajectory of 3x+1 sequence starting at 1.
%C A153727 From _Klaus Brockhaus_, May 23 2010: (Start)
%C A153727 Continued fraction expansion of (7+sqrt(229))/18.
%C A153727 Decimal expansion of 142/999. (End)
%C A153727 a(A008585(n)) = 1; a(A016777(n)) = 4; a(A016789(n)) = 2. - _Reinhard Zumkeller_, Oct 08 2011
%D A153727 C. A. Pickover, The Math Book, 2009; Collatz Conjecture, pp 374-375.
%H A153727 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>
%H A153727 Wikipedia, <a href="http://en.wikipedia.org/wiki/Collatz_conjecture">Collatz conjecture</a>
%H A153727 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%H A153727 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1).
%F A153727 a(3n)=1, a(3n+1)=4, a(3n+2)=2.
%F A153727 G.f.: (1+4*x+2*x^2)/(1-x^3).
%F A153727 a(n) = 4^n mod 7. - _Zerinvary Lajos_, Nov 25 2009
%F A153727 a(n) = A130196(n) * A131534(n). - _Reinhard Zumkeller_, Nov 12 2009
%F A153727 a(n) = 2^(-n mod 3) = 2^A080425(n). - _Wesley Ivan Hurt_, Jun 20 2014
%F A153727 a(n) = sqrt(4^(5*n) mod 21). - _Gary Detlefs_, Jul 07 2014
%F A153727 From _Wesley Ivan Hurt_, Jun 30 2016: (Start)
%F A153727 a(n) = a(n-3) for n>2.
%F A153727 a(n) = (7 - 4*cos(2*n*Pi/3) + 2*sqrt(3)*sin(2*n*Pi/3))/3. (End)
%p A153727 A153727:=n->2^(-n mod 3); seq(A153727(n), n=0..50); # _Wesley Ivan Hurt_, Jun 20 2014
%t A153727 a[n_] := {1, 4, 2}[[Mod[n, 3] + 1]]; Table[a[n], {n, 0, 104}] (* _Jean-François Alcover_, Jul 19 2013 *)
%t A153727 LinearRecurrence[{0, 0, 1},{1, 4, 2},105] (* _Ray Chandler_, Aug 25 2015 *)
%t A153727 PadRight[{},120,{1,4,2}] (* _Harvey P. Dale_, Jul 20 2025 *)
%o A153727 (Sage) [power_mod(2,-n,7)for n in range(0, 105)] # _Zerinvary Lajos_, Jun 07 2009
%o A153727 (Sage) [power_mod(4, n, 7)for n in range(0, 105)] # _Zerinvary Lajos_, Nov 25 2009
%o A153727 (PARI) A153727(n)=[1,4,2][n%3+1] \\ _M. F. Hasler_, Feb 10 2011
%o A153727 (Haskell)
%o A153727 a153727 n = a153727_list !! n
%o A153727 a153727_list = iterate a006370 1 -- _Reinhard Zumkeller_, Oct 08 2011
%o A153727 (Magma) [2^(-n mod 3) : n in [0..50]]; // _Wesley Ivan Hurt_, Jun 20 2014
%Y A153727 Row 1 of A347270.
%Y A153727 Cf. A178236 (decimal expansion of (7+sqrt(229))/18). Appears in A179133 (n>=1).
%Y A153727 Cf. A006370, A080425, A130196, A131534.
%K A153727 nonn,easy
%O A153727 0,2
%A A153727 _Philippe Deléham_, Dec 30 2008