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

%I A188510 #64 Feb 16 2025 08:33:14
%S A188510 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,
%T A188510 -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,
%U A188510 -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
%N A188510 Expansion of x*(1 + x^2)/(1 + x^4) in powers of x.
%H A188510 G. C. Greubel, <a href="/A188510/b188510.txt">Table of n, a(n) for n = 0..2500</a>
%H A188510 Michael Somos, <a href="http://cis.csuohio.edu/~somos/rfmc.html">Rational Function Multiplicative Coefficients</a>
%H A188510 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KroneckerSymbol.html">Kronecker Symbol.</a>
%H A188510 Wikipedia, <a href="https://en.wikipedia.org/wiki/Kronecker_symbol">Kronecker Symbol.</a>
%H A188510 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,-1).
%F A188510 Euler transform of length 8 sequence [0, 1, 0, -2, 0, 0, 0, 1].
%F A188510 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).
%F A188510 G.f.: x * (1 - x^4)^2/((1 - x^2)*(1 - x^8)) = (x + x^3)/(1 + x^4).
%F A188510 a(-n) = -a(n) = a(n+4).
%F A188510 a(n+2) = A091337(n).
%F A188510 a(2*n) = 0, a(2*n+1) = A057077(n).
%F A188510 G.f.: x/(1 - x^2/(1 + 2*x^2/(1 - x^2))). - _Michael Somos_, Jan 03 2013
%F A188510 a(n) = ((-2)/n), where (k/n) is the Kronecker symbol. Period 8. See the Eric Weisstein link. - _Wolfdieter Lang_, May 29 2013
%F A188510 a(n) = A257170(n) unless n = 0.
%F A188510 From _Jianing Song_, Nov 14 2018: (Start)
%F A188510 a(n) = sqrt(2)*sin(Pi*n/2)*cos(Pi*n/4).
%F A188510 E.g.f.: sqrt(2)*sin(x/sqrt(2))*cosh(x/sqrt(2)).
%F A188510 Moebius transform of A002325.
%F A188510 a(n) = A091337(n)*A101455(n).
%F A188510 a(n) = ((-2)^(2*i+1)/n) for all integers i >= 0, where (k/n) is the Kronecker symbol. (End)
%F A188510 a(n) = A014017(n-1)+A014017(n-3). - _R. J. Mathar_, Dec 17 2024
%e A188510 G.f. = x + x^3 - x^5 - x^7 + x^9 + x^11 - x^13 - x^15 + x^17 + x^19 - x^21 + ...
%t A188510 Table[KroneckerSymbol[-2, n], {n, 0, 104}] (* _Wolfdieter Lang_, May 30 2013 *)
%t A188510 a[ n_] := Mod[n, 2] (-1)^Quotient[ n, 4]; (* _Michael Somos_, Apr 17 2015 *)
%t A188510 CoefficientList[Series[x*(1+x^2)/(1+x^4), {x, 0, 60}], x] (* _G. C. Greubel_, Aug 02 2018 *)
%t A188510 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 *)
%o A188510 (PARI) {a(n) = (n%2) * (-1)^(n\4)};
%o A188510 (PARI) x='x+O('x^60); concat([0], Vec(x*(1+x^2)/(1+x^4))) \\ _G. C. Greubel_, Aug 02 2018
%o A188510 (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
%Y A188510 Cf. A002325, A057077, A091337, A101455, A257170.
%K A188510 sign,easy,mult
%O A188510 0,1
%A A188510 _Michael Somos_, Apr 10 2011