A163805 Expansion of (1 - x) * (1 - x^6) / ((1 - x^3) * (1 - x^4)) in powers of x.

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

Offset

0,1

Links

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

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

Formula

Euler transform of length 6 sequence [ -1, 0, 1, 1, 0, -1].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (2 - v) - u * (2 - u) * (3 - 2*v).

a(2*n) = 0 unless n=0, a(4*n + 1) = -1, a(4*n + 3) = a(0) = 1.

a(-n) = -a(n) unless n=0. a(n+4) = a(n) unless n=0 or n=-4.

a(n) = - A117569(n) unless n=0. a(n) = (-1)^n * A117569(n).

Convolution inverse of A163806.

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

G.f. A(x) = 1 - x / (1 + x^2) = 1 / (1 + x / (1 - x / (1 + x / (1 - x)))). - Michael Somos, Jan 03 2013

a(n) = A101455(n-2) = A056594(n-3), n>2. - R. J. Mathar, Aug 06 2009

E.g.f.: 1 - sin(x). - Stefano Spezia, Nov 16 2024

a(n) = sin(3*n*Pi/2), for n>0. - Ridouane Oudra, Nov 18 2024

Example

G.f. = 1 - x + x^3 - x^5 + x^7 - x^9 + x^11 - x^13 + x^15 - x^17 + x^19 + ...

Maple

1, seq(sin(3*n*Pi/2), n=1..100); # Ridouane Oudra, Nov 18 2024

Mathematica

a[ n_] := Boole[n == 0] + {-1, 0, 1, 0}[[Mod[n, 4, 1]]]; (* Michael Somos, Sep 06 2015 *)

Prog

(PARI) {a(n) = (n==0) + [0, -1, 0, 1][n%4 + 1]};
(PARI) {a(n) = (n==0) - kronecker(-4, n)};

Crossrefs

Cf. A117569, A163806.

Sequence in context: A073097 A117569 A135528 * A267015 A078387 A189292

Adjacent sequences: A163802 A163803 A163804 * A163806 A163807 A163808

Keyword

sign,easy

Author

Michael Somos, Aug 04 2009

Status

approved