A167817 Period 4: repeat [1, 3, 3, 3].

1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1

Offset: 0

Author

Reinhard Zumkeller, Nov 13 2009

Keywords

• frac
• nonn
• easy

Comments

Denominator of x(n) = x(n-1) + x(n-2), x(0)=0, x(1)=1/3; numerator = A167816(n).

Continued fraction expansion of (33 + sqrt(2805))/66. - Klaus Brockhaus, May 06 2010

Links

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

Crossrefs

Cf. A130196.

Cf. A177344 (decimal expansion of (33+sqrt(2805))/66). - Klaus Brockhaus, May 06 2010

Programs

Magma

&cat[[1, 3, 3, 3]: n in [0..50]]; // Vincenzo Librandi, Dec 28 2010

Maple

seq(op([1, 3, 3, 3]), n=0..50); # Wesley Ivan Hurt, Jul 09 2016

Mathematica

Denominator[LinearRecurrence[{1,1},{0,1/3},110]] (* or *) PadRight[{},110,{1,3,3,3}] (* Harvey P. Dale, Dec 07 2014 *)
LinearRecurrence[{0, 0, 0, 1},{1, 3, 3, 3},105] (* Ray Chandler, Aug 03 2015 *)

Formula

a(n) = 3 - 2 * 0^(n mod 4).

G.f.: (1 + 3*x + 3*x^2 + 3*x^3)/(1-x^4). - Klaus Brockhaus, May 06 2010

a(n) = 5/2 - cos(Pi*n/2) - (-1)^n/2. - R. J. Mathar, Oct 08 2011

E.g.f.: -cos(x) + 3*sinh(x) + 2*cosh(x). - Ilya Gutkovskiy, Jun 27 2016

a(n) = a(n-4) for n>3. - Wesley Ivan Hurt, Jul 09 2016

Extensions

Definition corrected by D. S. McNeil, May 09 2010