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 A027009 #9 Jul 15 2025 17:13:15
%S A027009 1,1,9,14,46,81,209,389,901,1726,3774,7349,15541,30561,63329,125294,
%T A027009 256366,509161,1033449,2057549,4154701,8284926,16673534,33282989,
%U A027009 66837421,133507081,267724809,535010414,1071881326,2142612801,4290096449,8577182549,17167117141,34326353086,68686091454,137351549669,274790503141
%N A027009 a(n) = Sum_{k=floor((n+2)/2)..n} T(n, k), T given by A026998.
%H A027009 G. C. Greubel, <a href="/A027009/b027009.txt">Table of n, a(n) for n = 1..1000</a>
%H A027009 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-6,-1,2).
%F A027009 G.f.: x*(1-x+4*x^2-x^3-2*x^4)/((1-2*x)*(1+x-x^2)*(1-x-x^2)).
%F A027009 From _G. C. Greubel_, Jul 11 2025: (Start)
%F A027009 a(n) = 2^(n+1) - (1/2)*(A000032(n+3) + (-1)^n*A000032(n)).
%F A027009 E.g.f.: 1 + 2*exp(2*x) - 2*cosh(x/2)*cosh(sqrt(5)*x/2) - exp(x/2)*(cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2)). (End)
%t A027009 LinearRecurrence[{2,3,-6,-1,2}, {1,1,9,14,46}, 41] (* _G. C. Greubel_, Jul 11 2025 *)
%o A027009 (Magma)
%o A027009 A027009:= func< n | 2^(n+1) -(Lucas(n+3) +(-1)^n*Lucas(n))/2 >;
%o A027009 [A027009(n): n in [1..40]]; // _G. C. Greubel_, Jul 11 2025
%o A027009 (SageMath)
%o A027009 def lucas(n): return lucas_number2(n,1,-1)
%o A027009 def A027009(n): return 2^(n+1) -(lucas(n+3) +(-1)^n*lucas(n))//2
%o A027009 print([A027009(n) for n in range(1,41)]) # _G. C. Greubel_, Jul 11 2025
%Y A027009 Cf. A000032, A026998.
%K A027009 nonn
%O A027009 1,3
%A A027009 _Clark Kimberling_
%E A027009 More terms added by _G. C. Greubel_, Jul 11 2025