This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A099489 #20 Jan 15 2025 03:00:38
%S A099489 1,3,11,42,157,585,2183,8148,30409,113487,423539,1580670,5899141,
%T A099489 22015893,82164431,306641832,1144402897,4270969755,15939476123,
%U A099489 59486934738,222008262829,828546116577,3092176203479,11540158697340
%N A099489 Expansion of (1-x+x^2)/((1+x^2)(1-4x+x^2)).
%C A099489 A Chebyshev transform of the sequence A002001 which has with g.f. (1-x)/(1-4x). The image of G(x) under the Chebyshev transform is (1/(1+x^2))G(x/(1+x^2)).
%H A099489 Matthew House, <a href="/A099489/b099489.txt">Table of n, a(n) for n = 0..1739</a>
%H A099489 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,4,-1).
%F A099489 a(n) = 4*a(n-1)-2*a(n-2)+4*a(n-3)-a(n-4). - corrected by _Matthew House_, Oct 22 2016
%F A099489 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*(3*4^(n-2*k)+0^(n-2*k))/4.
%F A099489 a(n) = Sum_{k=0..n} (0^k-sin(Pi*k/2))*((2+sqrt(3))^(n-k+1)-(2-sqrt(3))^(n-k+1))/(2*sqrt(3)).
%F A099489 a(n) = Sum_{k=0..n} (0^k-sin(Pi*k/2))*A001353(n-k+1).
%F A099489 a(n) = 3*A001353(n+1)/4 +A056594(n)/4. - _R. J. Mathar_, Sep 21 2012
%t A099489 CoefficientList[Series[(1-x+x^2)/((1+x^2)(1-4x+x^2)),{x,0,30}],x] (* or *_)
%t A099489 LinearRecurrence[{4,-2,4,-1},{1,3,11,42},30] (* _Harvey P. Dale_, Dec 28 2019 *)
%Y A099489 Cf. A099486, A099487, A099488.
%K A099489 easy,nonn
%O A099489 0,2
%A A099489 _Paul Barry_, Oct 18 2004