A110514 Expansion of (1 - x + x^2 + x^3)/(1 - x^2 - x^4 + x^6).

1, -1, 2, 0, 3, -1, 4, 0, 5, -1, 6, 0, 7, -1, 8, 0, 9, -1, 10, 0, 11, -1, 12, 0, 13, -1, 14, 0, 15, -1, 16, 0, 17, -1, 18, 0, 19, -1, 20, 0, 21, -1, 22, 0, 23, -1, 24, 0, 25, -1, 26, 0, 27, -1, 28, 0, 29, -1, 30, 0, 31, -1, 32, 0, 33, -1, 34, 0, 35, -1, 36, 0, 37, -1, 38, 0, 39, -1, 40, 0, 41, -1, 42, 0, 43, -1, 44, 0, 45, -1, 46, 0, 47, -1, 48, 0

Offset

0,3

Comments

Diagonal sums of A110515. Partial sums of A110516.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Formula

G.f.: (1 - x + x^2 + x^3)/((1 - x^2)^2(1 + x^2)).

a(n) = a(n-2) + a(n-4) - a(n-6).

a(n) = (n/2 + 1)*(1 + (-1)^n)/2 - (1 - (-1)^n)*(1 + (-1)^((n-1)/2))/4.

a(n) = (sin(Pi*n/2)*((-1)^n - 1) + (n+3)*(-1)^n + (n+1))/4.

a(n) = Sum_{k=0..floor(n/2)} Jacobi(2^(n-2k), 2(n-2k)+1) [conjecture].

Mathematica

Riffle[Range[50], {-1, 0}] (* Harvey P. Dale, Dec 08 2011 *)

Prog

(PARI) x='x+O('x^50); Vec((1-x+x^2+x^3)/((1-x^2)^2(1+x^2))) \\ G. C. Greubel, Aug 29 2017

Crossrefs

Cf. A106249.

Sequence in context: A249738 A364020 A363898 * A338503 A249122 A135157

Adjacent sequences: A110511 A110512 A110513 * A110515 A110516 A110517

Keyword

easy,sign

Author

Paul Barry, Jul 24 2005

Status

approved