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 A330051 #29 Apr 19 2023 00:40:58
%S A330051 0,0,2,7,25,72,208,564,1530,4059,10769,28336,74560,195576,513010,
%T A330051 1344063,3521385,9221688,24149456,63230860,165558250,433454835,
%U A330051 1134845857,2971111392,7778592000,20364739632,53315898338,139583151799,365434267705,956720165544
%N A330051 a(n) = 1 + F(2*n+1) - (F(n+4) - (-1)^n*F(n-2))/2 where F=A000045.
%H A330051 Vaclav Kotesovec, <a href="https://mathematica.stackexchange.com/q/206655">Why is this product equal to zero, when the correct result is 2+GoldenRatio</a>, Mathematica StackExchange, Sep 22 2019.
%H A330051 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1,-11,11,1,-4,1).
%F A330051 a(n) = 1 + F(2*n+1) - F(n+2) - (F(-n+2) + F(n+1))/2.
%F A330051 G.f.: (2*x^2 - x^3 - x^4 + x^5) / (1 - 4*x + x^2 + 11*x^3 - 11*x^4 - x^5 + 4*x^6 - x^7).
%F A330051 b(n) + a(n) * sqrt(5) = F(2*n+2) * Product_{k=2..n} 1 / (1 - q^k/(1 - q^(2*k))) where q = (sqrt(5)-1)/2 and b=A330050.
%F A330051 a(n) = A005013(floor(n/2)) * A329421(n).
%e A330051 G.f. = 2*x^2 + 7*x^3 + 25*x^4 + 72*x^5 + 208*x^6 + 564*x^7 + 1530*x^8 + ...
%t A330051 a[n_] := 1 + Fibonacci[2 n + 1] - (Fibonacci[n + 4] - (-1)^n Fibonacci[n - 2])/2
%o A330051 (PARI) {a(n) = 1 + fibonacci(2*n + 1) - (fibonacci(n + 4) - (-1)^n*fibonacci(n - 2))/2};
%Y A330051 Cf. A000045, A005013, A329421, A330050.
%K A330051 nonn,easy
%O A330051 0,3
%A A330051 _Michael Somos_, Dec 01 2019
%E A330051 Definition corrected by _N. J. A. Sloane_, May 29 2022 following a suggestion from _Kevin Ryde_.
%E A330051 Additional corrections by _Eric Rowland_, May 31 2022