A188510 Expansion of x*(1 + x^2)/(1 + x^4) in powers of x.

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

Offset

0,1

Comments

From Jianing Song, Dec 13 2025: (Start)

The Dirichlet character associated with the imaginary quadratic field Q(sqrt(-2)) (discriminant -8).

Note that (Sum_{i=0..7} i*a(i))/(-8) = 1 gives the class number of the imaginary quadratic field Q(sqrt(-2)), i.e., the corresponding ring of integers Z[sqrt(-2)] is a unique factorization domain. (End)

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

Michael Somos, Rational Function Multiplicative Coefficients

Eric Weisstein's World of Mathematics, Class Number.

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(n+2) = A091337(n).

a(2*n) = 0, 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 8. See the Eric Weisstein link. - Wolfdieter Lang, May 29 2013

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

From Jianing Song, Nov 14 2018: (Start)

a(n) = sqrt(2)*sin(Pi*n/2)*cos(Pi*n/4).

E.g.f.: sqrt(2)*sin(x/sqrt(2))*cosh(x/sqrt(2)).

a(n) = A091337(n)*A101455(n).

a(n) = ((-2)^(2*i+1)/n) for all integers i >= 0, where (k/n) is the Kronecker symbol. (End)

a(n) = A014017(n-1)+A014017(n-3). - R. J. Mathar, Dec 17 2024

Sum_{n>=1} a(n)/n = -(Pi/8^(3/2)) * (Sum_{i=0..7} i*a(i)) = Pi/sqrt(8) (Dirichlet class number formula). - Jianing Song, Dec 13 2025

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 *)
CoefficientList[Series[x*(1+x^2)/(1+x^4), {x, 0, 60}], x] (* G. C. Greubel, Aug 02 2018 *)
LinearRecurrence[{0, 0, 0, -1}, {0, 1, 0, 1}, 120] (* or *) PadRight[{}, 120, {0, 1, 0, 1, 0, -1, 0, -1}] (* Harvey P. Dale, Jan 25 2023 *)

Prog

(PARI) {a(n) = (n%2) * (-1)^(n\4)};
(PARI) x='x+O('x^60); concat([0], Vec(x*(1+x^2)/(1+x^4))) \\ G. C. Greubel, Aug 02 2018
(Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x^2)/(1+x^4))); // G. C. Greubel, Aug 02 2018

Crossrefs

Cf. A057077, A257170.

Moebius transform of A002325.

Cf. A033203 (primes not inert in Q(sqrt(-2))), A033200 (primes decomposing), A003628 (primes remaining inert), A045355 (primes not decomposing).

Kronecker symbols {(D/n)} for negative fundamental discriminants D = -3..-47, -67, -163: A102283, A101455, A175629, this sequence, A011582, A316569, A011585, A289741, A011586, A109017, A011588, A390614, A388073, A388072, A011591, A011592, A011596, A011615.

Kronecker symbols {(D/n)} for positive fundamental discriminants D = 5..41: A080891, A091337, A110161, A011583, A011584, A322829, A322796, A390615, A011587, A391502, A011589, A391503, A011590.

Sequence in context: A379964 A101455 A000035 * A091337 A131734 A134452

Adjacent sequences: A188507 A188508 A188509 * A188511 A188512 A188513

Keyword

sign,easy,mult

Author

Michael Somos, Apr 10 2011

Status

approved