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URL: https://oeis.org/A188510

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A188510
Expansion of x*(1 + x^2)/(1 + x^4) in powers of x.
42
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
Eric Weisstein's World of Mathematics, Class Number.
Eric Weisstein's World of Mathematics, Kronecker Symbol.
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
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
KEYWORD
sign,easy,mult
AUTHOR
Michael Somos, Apr 10 2011
STATUS
approved