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 A022408 #23 Sep 13 2024 19:49:46
%S A022408 3,9,13,23,37,61,99,161,261,423,685,1109,1795,2905,4701,7607,12309,
%T A022408 19917,32227,52145,84373,136519,220893,357413,578307,935721,1514029,
%U A022408 2449751,3963781,6413533,10377315,16790849,27168165,43959015,71127181,115086197,186213379,301299577,487512957
%N A022408 a(n) = a(n-1) + a(n-2) + 1, with a(0)=3, a(1)=9.
%H A022408 G. C. Greubel, <a href="/A022408/b022408.txt">Table of n, a(n) for n = 0..1000</a>
%H A022408 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1)
%F A022408 From _R. J. Mathar_, Mar 11 2011: (Start)
%F A022408 a(n+1) - a(n) = A022382(n-1).
%F A022408 G.f.: ( 3+3*x-5*x^2 ) / ( (x-1)*(x^2+x-1) ). (End)
%F A022408 a(n) = 2*Lucas(n+1) + 2*Fibonacci(n+2) - 1. - _Greg Dresden_, Oct 10 2020
%t A022408 CoefficientList[Series[(3+3*x-5*x^2)/((x-1)*(x^2+x-1)), {x, 0, 50}], x] (* or *) LinearRecurrence[{2,0,-1}, {3,9,13}, 50] (* _G. C. Greubel_, Mar 01 2018 *)
%t A022408 nxt[{a_,b_}]:={b,a+b+1}; NestList[nxt,{3,9},40][[;;,1]] (* _Harvey P. Dale_, Sep 13 2024 *)
%o A022408 (PARI) x='x+O('x^40); Vec((3+3*x-5*x^2)/((x-1)*(x^2+x-1))) \\ _G. C. Greubel_, Mar 01 2018
%o A022408 (Magma) I:=[3,9,13]; [n le 3 select I[n] else 2*Self(n-1) - Self(n-3): n in [1..40]]; // _G. C. Greubel_, Mar 01 2018
%Y A022408 Cf. A022382.
%K A022408 nonn
%O A022408 0,1
%A A022408 _N. J. A. Sloane_
%E A022408 Terms a(31) onward added by _G. C. Greubel_, Mar 01 2018