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A163806 Expansion of (1 - x^3) * (1 - x^4) / ((1 - x) * (1 - x^6)) in powers of x. 3
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 (list; graph; refs; listen; history; text; internal format)
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 (1, -1).

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.

EXAMPLE

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

MATHEMATICA

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

PROG

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

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

(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

CROSSREFS

Cf. A128834, A163804, A163805, A163810.

Sequence in context: A131309 A267208 A106510 * A163810 A163804 A181653

Adjacent sequences:  A163803 A163804 A163805 * A163807 A163808 A163809

KEYWORD

sign,easy,mult

AUTHOR

Michael Somos, Aug 04 2009

STATUS

approved

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Last modified October 19 22:55 EDT 2019. Contains 328244 sequences. (Running on oeis4.)