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 A174542 #28 Sep 05 2023 11:18:47
%S A174542 1,2,5,10,23,44,99,188,421,798,1785,3382,7563,14328,32039,60696,
%T A174542 135721,257114,574925,1089154,2435423,4613732,10316619,19544084,
%U A174542 43701901,82790070,185124225,350704366,784198803,1485607536,3321919439,6293134512,14071876561
%N A174542 Partial sums of odd Fibonacci numbers (A014437).
%H A174542 Colin Barker, <a href="/A174542/b174542.txt">Table of n, a(n) for n = 1..1000</a>
%H A174542 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-4,1,-1).
%F A174542 a(2n) = (1/2)*(Fibonacci(3n+2)-1).
%F A174542 a(2n+1) = (1/2)*(Fibonacci(3n+1)+Fibonacci(3n+3)-1).
%F A174542 a(n) = a(n-1)+4*a(n-2)-4*a(n-3)+a(n-4)-a(n-5) for n>4. - _Colin Barker_, Oct 26 2014
%F A174542 G.f.: x*(x^3-x^2+x+1) / ((x-1)*(x^4+4*x^2-1)). - _Colin Barker_, Oct 26 2014
%F A174542 From _Vladimir Reshetnikov_, Oct 30 2015: (Start)
%F A174542 a(n) = ((sin(Pi*n/2)*sqrt(5/phi) - cos(Pi*n/2)/phi^2)/phi^(3*n/2) + (sqrt(5*phi)*sin(Pi*n/2)^2 + phi^2*cos(Pi*n/2)^2)*phi^(3*n/2))/(2*sqrt(5)) - 1/2, where phi=(1+sqrt(5))/2.
%F A174542 E.g.f.: phi^2*cosh(phi^(3/2)*x)/(2*sqrt(5)) + sqrt(phi)*sinh(phi^(3/2)*x)/2 - cos(x/phi^(3/2))/(2*sqrt(5)*phi^2) + sin(x/phi^(3/2))/(2*sqrt(phi)) - exp(x)/2.
%F A174542 (End)
%t A174542 s=0;lst={};Do[f=Fibonacci[n];If[OddQ[f],AppendTo[lst,s+=f]],{n,0,5!}];lst
%t A174542 CoefficientList[Series[(x^3 - x^2 + x + 1)/((x - 1) (x^4 + 4 x^2 - 1)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Oct 27 2014 *)
%t A174542 Accumulate[Select[Fibonacci[Range[50]],OddQ]] (* or *) LinearRecurrence[{1,4,-4,1,-1},{1,2,5,10,23},40] (* _Harvey P. Dale_, Sep 05 2023 *)
%o A174542 (PARI) Vec(x*(x^3-x^2+x+1)/((x-1)*(x^4+4*x^2-1)) + O(x^100)) \\ _Colin Barker_, Oct 26 2014
%o A174542 (Magma) &cat[[(1/2)*(Fibonacci(3*n+2)-1), (1/2)*(Fibonacci(3*n+1)+Fibonacci(3*n+3)-1)]: n in [1..30]]; // _Vincenzo Librandi_, Oct 27 2014
%Y A174542 Cf. A000045, A099919, A014437.
%K A174542 nonn,easy
%O A174542 1,2
%A A174542 _Vladimir Joseph Stephan Orlovsky_, Nov 28 2010