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 A106511 #37 Oct 19 2023 17:00:39
%S A106511 1,0,0,0,1,-2,3,-4,6,-10,17,-28,45,-72,116,-188,305,-494,799,-1292,
%T A106511 2090,-3382,5473,-8856,14329,-23184,37512,-60696,98209,-158906,257115,
%U A106511 -416020,673134,-1089154,1762289,-2851444,4613733,-7465176,12078908,-19544084,31622993
%N A106511 Expansion of g.f. (1+x)^2/((1 + x + x^2)*(1 + x - x^2)).
%C A106511 Diagonal sums of the Riordan array ((1+x)/(1+x+x^2), x/(1+x)), A106509.
%H A106511 Vincenzo Librandi, <a href="/A106511/b106511.txt">Table of n, a(n) for n = 0..1000</a>
%H A106511 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-2,-1,0,1).
%F A106511 a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-2k} (-1)^j*binomial(2n-3k-j, j).
%F A106511 a(n) = (1/2)*((-1)^n*Fibonacci(n) + Kronecker(-3,n)). - _Ralf Stephan_, Jun 02 2007
%F A106511 a(n) = -2*a(n-1) - a(n-2) + a(n-4), a(0)=1, a(1)=a(2)=a(3)=0. - _Philippe Deléham_, Jan 12 2014
%F A106511 a(n) = (-1)^n*(Fibonacci(n-1) - A005252(n-2)), n>=2. - _Katharine Ahrens_, May 05 2019
%F A106511 E.g.f.: exp(-x/2)*(15*cos(sqrt(3)*x/2) + 15*cosh(sqrt(5)*x/2) + 5*sqrt(3)*sin(sqrt(3)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/30. - _Stefano Spezia_, Oct 19 2023
%t A106511 CoefficientList[Series[(1 + x)^2/((1 + x + x^2)(1 + x - x^2)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jan 12 2014 *)
%o A106511 (Magma) I:=[1,0,0,0]; [n le 4 select I[n] else -2*Self(n-1)-Self(n-2)+Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Jan 12 2014
%o A106511 (Sage)
%o A106511 def A005252(n): return sum( binomial(n-2*k, 2*k) for k in (0..n//4) )
%o A106511 def A106511(n): return (-1)^n*( fibonacci(n-1) - A005252(n-2) )
%o A106511 [A106511(n) for n in (0..45)] # _G. C. Greubel_, Apr 29 2021
%o A106511 (PARI) a(n) = (fibonacci(1-n) + 1 - n%3) >> 1; \\ _Kevin Ryde_, Apr 29 2021
%Y A106511 Cf. A000045, A005252, A011646, A024490, A039834, A106509.
%K A106511 easy,sign
%O A106511 0,6
%A A106511 _Paul Barry_, May 04 2005