A101675 Expansion of (1 - x - x^2)/(1 + x^2 + x^4).

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

Offset: 0

Author

Paul Barry, Dec 11 2004

Keywords

• easy
• sign

Comments

Partial sums are A101676.

Periodic with period 6. - Ray Chandler, Sep 03 2015

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (0,-1,0,-1).

Crossrefs

Cf. A101676.

Programs

Magma

I:=[1,-1,-2,1]; [n le 4 select I[n] else -Self(n-2)-Self(n-4): n in [1..120]]; // Vincenzo Librandi, Sep 04 2015

Mathematica

LinearRecurrence[{0, -1, 0, -1},{1, -1, -2, 1},105] (* Ray Chandler, Sep 03 2015 *)
CoefficientList[Series[(1 - x - x^2)/(1 + x^2 + x^4), {x, 0, 150}], x] (* Vincenzo Librandi, Sep 04 2015 *)

PARI

Vec((1-x-x^2)/(1+x^2+x^4) + O(x^80)) \\ Michel Marcus, Sep 04 2015

Formula

a(0) = 1, a(1) = -1, a(2) = -2, a(3) = 1; for n >= 4, a(n) = -a(n-2)-a(n-4).

a(n) = Sum_{k=0..floor(n/2)} (-1)^A010060(n-2k)*(binomial(n-k, k) mod 2)*(-1)^k.

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

a(n) = (-1)^(n+1)*H(n + 4, n mod 2, 1/2) where H(n, a, b) = hypergeom([a - n/2, b - n/2], [1 - n], 4). - Peter Luschny, Sep 03 2019