A166445 - OEIS

%I #38 Jul 28 2024 00:14:09

%S 1,0,-1,1,3,0,-3,1,5,0,-5,1,7,0,-7,1,9,0,-9,1,11,0,-11,1,13,0,-13,1,

%T 15,0,-15,1,17,0,-17,1,19,0,-19,1,21,0,-21,1,23,0,-23,1,25,0,-25,1,27,

%U 0,-27,1,29,0,-29,1,31,0,-31,1,33,0,-33,1,35,0,-35,1

%N Hankel transform of A025276.

%H Felix Fröhlich, <a href="/A166445/b166445.txt">Table of n, a(n) for n = 0..10000</a>

%H Han Wang and Zhi-Wei Sun, <a href="https://arxiv.org/abs/2206.12317">Evaluations of three determinants</a>, arXiv:2206.12317 [math.NT], 2022.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,-2,2,-1,1).

%F G.f.: (1-x+x^2+x^4)/((1-x)*(1+x^2)^2).

%F a(n) = (1/2)*(1 + cos((n+1)*Pi/2) + (n+1)*sin((n+1)*Pi/2)). - _Harvey P. Dale_, Nov 21 2014; corrected by _Bernard Schott_, Jun 27 2022

%F For n >= 0: a(4n) = 2n+1; a(4n+1) = 0; a(4n+2) = -a(4n) = -2n-1; a(4n+3) = 1. - _Bernard Schott_, Jun 27 2022

%F a(n) - a(n-1) = A127365(n+1). - _R. J. Mathar_, Jul 01 2024

%F E.g.f.: (exp(x) + cos(x) - (1 + x)*sin(x))/2. - _Stefano Spezia_, Jul 14 2024

%F a(n) = (1/2)*(1 - A056594(n) - A056594(n-1) + 2*(-1)^floor(n/2) * A027656(n)). - _G. C. Greubel_, Jul 27 2024

%t LinearRecurrence[{1,-2,2,-1,1},{1,0,-1,1,3},80] (* _Harvey P. Dale_, Nov 21 2014 *)

%o (PARI) Vec((1-x+x^2+x^4)/((1-x)*(1+x^2)^2) + O(x^80)) \\ _Felix Fröhlich_, Jun 28 2022

%o (Magma)

%o R<x>:=PowerSeriesRing(Integers(), 80);

%o Coefficients(R!( (1-x+x^2+x^4)/((1-x)*(1+x^2)^2) )); // _G. C. Greubel_, Jul 27 2024

%o (SageMath)

%o def A166445_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( (1-x+x^2+x^4)/((1-x)*(1+x^2)^2) ).list()

%o A166445_list(80) # _G. C. Greubel_, Jul 27 2024

%Y Cf. A025276, A027656, A056594, A127365.

%K easy,sign

%O 0,5

%A _Paul Barry_, Oct 13 2009