A117188 Expansion of (1-x^2)/(1+x^2+x^4).

1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0

Offset: 0

Author

Paul Barry, Mar 01 2006

Keywords

• sign
• easy

Comments

Periodic: repeat [1, 0, -2, 0, 1, 0].

Minton mentions that the subsequence a(2^i), i >= 1, oscillates between -2 and 1 (and does not converge 2-adically). - N. J. A. Sloane, Jul 09 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Gregory T. Minton, Linear recurrence sequences satisfying congruence conditions, Proc. Amer. Math. Soc. 142 (2014), no. 7, 2337--2352. MR3195758. See Example 6.13. - _N. J. A. Sloane_, Jul 09 2014
• Index entries for linear recurrences with constant coefficients, signature (0,-1,0,-1).

Crossrefs

Row sums of A117185.

Programs

Magma

&cat [[1, 0, -2, 0, 1, 0]^^20]; // Wesley Ivan Hurt, Jun 23 2016

Maple

A117188:=n->[1, 0, -2, 0, 1, 0][(n mod 6)+1]: seq(A117188(n), n=0..100); # Wesley Ivan Hurt, Jun 23 2016

Mathematica

PadRight[{}, 100, {1, 0, -2, 0, 1, 0}] (* Wesley Ivan Hurt, Jun 23 2016 *)
LinearRecurrence[{0,-1,0,-1},{1,0,-2,0},100] (* Harvey P. Dale, Jun 25 2022 *)

Formula

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

a(n) = (1 + (-1)^n)/(-2 + 4^(floor((n-1)/3) - 2*floor((n-1)/6))). - Tani Akinari, Aug 02 2013

a(n) = -a(n-2) - a(n-4) for n >= 4. - N. J. A. Sloane, Jul 09 2014

From Wesley Ivan Hurt, Jun 23 2016: (Start)

a(n) = a(n-6) for n>5.

a(n) = cos(n*Pi/2) * (cos(n*Pi/6) + sqrt(3)*sin(n*Pi/6)). (End)

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

a(n) = cos((n+1)*Pi/3) - cos(2*(n+1)*Pi/3). - Ridouane Oudra, Dec 14 2021