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A112299
Expansion of x * (1 - x) * (1 - x^2) * (1 - x^3) / (1 - x^8) in powers of x.
4
1, -1, -1, 0, 1, 1, -1, 0, 1, -1, -1, 0, 1, 1, -1, 0, 1, -1, -1, 0, 1, 1, -1, 0, 1, -1, -1, 0, 1, 1, -1, 0, 1, -1, -1, 0, 1, 1, -1, 0, 1, -1, -1, 0, 1, 1, -1, 0, 1, -1, -1, 0, 1, 1, -1, 0, 1, -1, -1, 0, 1, 1, -1, 0, 1, -1, -1, 0, 1, 1, -1, 0, 1, -1, -1, 0, 1, 1, -1, 0, 1, -1, -1, 0, 1, 1, -1, 0, 1, -1, -1, 0, 1, 1, -1, 0, 1, -1, -1, 0, 1, 1, -1, 0, 1
OFFSET
1,1
COMMENTS
Periodic with period length 8.
Sum_{k>=1} a(k)/k = Pi/8. - Jaume Oliver Lafont, Oct 20 2009
FORMULA
Euler transform of length 8 sequence [-1, -1, -1, 0, 0, 0, 0, 1].
Multiplicative with a(2) = -1, a(2^e) = 0 if e>1, a(p^e) = 1 if p == 1 (mod 4), a(p^e) = (-1)^e if p == 3 (mod 4).
G.f.: x * (1 + x + x^2) * (1 - x)^2 / ((1 + x^2) * (1 + x^4)).
G.f.: f(x) - f(x^2) where f(x) := x / (1 + x^2). - Michael Somos, Jun 19 2015
a(n) = -a(4 - n) = a(n + 8) for all n in Z. a(4*n) = 0.
a(2*n) = - A056594(n-1). a(2*n + 1) = A033999(n). a(4*n + 1) = 1. a(4*n + 3) = -1. a(4*n + 2) = - A033999(n). - Michael Somos, Jun 19 2015
a(n) = A257196(n) unless n=0. - Michael Somos, Sep 01 2015
EXAMPLE
G.f. = x - x^2 - x^3 + x^5 + x^6 - x^7 + x^9 - x^10 - x^11 + x^13 + x^14 - x^15 + ...
MATHEMATICA
LinearRecurrence[{0, -1, 0, -1, 0, -1}, {1, -1, -1, 0, 1, 1}, 110] (* Harvey P. Dale, Dec 07 2014 *)
a[ n_] := {1, -1, -1, 0, 1, 1, -1, 0}[[Mod[n, 8, 1]]];
PROG
(PARI) {a(n) = [0, 1, -1, -1, 0, 1, 1, -1][n%8 + 1]};
(PARI) {a(n) = [0, 1, -(-1)^(n\4), -1][n%4 + 1]}; /* Michael Somos, Jun 19 2015 */
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+x+x^2)*(1-x)^2/((1+x^2)*(1+x^4)))); // G. C. Greubel, Aug 03 2018
CROSSREFS
KEYWORD
sign,mult,easy
AUTHOR
Michael Somos, Sep 02 2005
STATUS
approved