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A188510 Expansion of x*(1 + x^2) / (1 + x^4) in powers of x. 2
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, -1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

LINKS

Table of n, a(n) for n=0..104.

M. Somos, Rational Function Multiplicative Coefficients

Eric Weisstein's World of Mathematics, Kronecker Symbol.

Wikipedia, Kronecker Symbol.

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

FORMULA

Euler transform of length 8 sequence [ 0, 1, 0, -2, 0, 0, 0, 1].

a(n) is multiplicative with a(2^e) = 0^e, a(p^e) = 1 if p == 1 or 3 (mod 8), a(p^e) = (-1)^e if p == 5 or 7 (mod 8).

G.f.: x * (1 - x^4)^2 / ((1 - x^2) * (1 - x^8)) = (x + x^3) / (1 + x^4). a(-n) = -a(n) = a(n + 4). a(2*n) = 0.

a(n + 2) = A091337(n). a(2*n + 1) = A057077(n).

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

a(n) = (-2,n), where (k/n) is the Kronecker symbol. Period length 8. See the Eric Weisstein link. - Wolfdieter Lang, May 29 2013

a(n) = A257170(n) unless n = 0.

EXAMPLE

G.f. = x + x^3 - x^5 - x^7 + x^9 + x^11 - x^13 - x^15 + x^17 + x^19 - x^21 + ...

MATHEMATICA

Table[KroneckerSymbol[-2, n], {n, 0, 104}]. - Wolfdieter Lang, May 30 2013

a[ n_] := Mod[n, 2] (-1)^Quotient[ n, 4]; (* Michael Somos, Apr 17 2015 *)

PROG

(PARI) {a(n) = (n%2) * (-1)^(n\4)};

CROSSREFS

Cf. A057077, A091337, A257170.

Sequence in context: A173923 A125122 A000035 * A131734 A134452 A073445

Adjacent sequences:  A188507 A188508 A188509 * A188511 A188512 A188513

KEYWORD

sign,easy,mult

AUTHOR

Michael Somos, Apr 10 2011

STATUS

approved

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Last modified February 18 15:08 EST 2018. Contains 299324 sequences. (Running on oeis4.)