A163806 Expansion of (1 - x^3) * (1 - x^4) / ((1 - x) * (1 - x^6)) in powers of x.

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

Offset: 0

Author

Michael Somos, Aug 04 2009

Keywords

• sign
• easy
• mult

Examples

			G.f. = 1 + x + x^2 - x^4 - x^5 + x^7 + x^8 - x^10 - x^11 + x^13 + x^14 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,-1).

Crossrefs

Cf. A128834, A163804, A163805, A163810.

Programs

Magma

I:=[1,1]; [1] cat [n le 2 select I[n] else Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 15 2018

Mathematica

Join[{1},LinearRecurrence[{1, -1},{1, 1},104]] (* Ray Chandler, Sep 15 2015 *)

PARI

{a(n) = (n==0) + [0, 1, 1, 0, -1, -1][n%6 + 1]};

PARI

{a(n) = (n==0) - (-1)^n * kronecker(-3, n)};

Formula

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

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * (2*v - 1) - (3*v - 2) * (2*u - 1).

a(n) is multiplicative with a(3^e) = 0^e, a(2^e) = -(-1)^e if e>0, a(p^e) = 1 if p == 1 (mod 6), a(p^e) = (-1)^e if p == 5 (mod 6).

a(3*n) = 0 unless n=0, a(6*n + 4) = a(6*n + 5) = -1, a(6*n + 1) = a(6*n + 2) = a(0) = 1.

a(-n) = -a(n) unless n=0. a(n+3) = -a(n) unless n=0 or n=-3.

G.f.: (1 + x^2) / (1 - x + x^2). - Corrected by Bruno Berselli, Apr 06 2011

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

A128834(n) = a(n) unless n=0. A163810(n) = -a(n) unless n=0. A163804(n) = (-1)^n * a(n). Convolution inverse of A163805.