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

Author

Michael Somos, Apr 10 2011

Keywords

• sign
• easy
• mult

Examples

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

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

Crossrefs

Cf. A002325, A057077, A091337, A101455, A257170.

Programs

Magma

m:=60; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x^2)/(1+x^4))); // G. C. Greubel, Aug 02 2018

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 *)

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

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)).

Moebius transform of A002325.

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