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 A193530 #26 Sep 08 2022 08:45:58
%S A193530 1,1,2,3,7,13,31,66,159,363,876,2065,4985,11915,28765,69156,166957,
%T A193530 402373,971414,2343519,5657755,13654969,32966011,79577190,192116331,
%U A193530 463786191,1119678912,2703086893,6525829037,15754607063,38034986041,91824246216,221683340569,535190123593,1292063254826
%N A193530 Expansion of (1 - 2*x - 2*x^2 + 3*x^3 + x^5)/((1-x)*(1-2*x-x^2)*(1-2*x^2-x^4)).
%C A193530 This sequence was initially confused with A003120, but they are different sequences. The g.f. used here as the definition was found by Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009.
%H A193530 Vincenzo Librandi, <a href="/A193530/b193530.txt">Table of n, a(n) for n = 0..1000</a>
%H A193530 Gy. Tasi and F. Mizukami, <a href="http://dx.doi.org/10.1023/A:1019163812482">Quantum algebraic-combinatoric study of the conformational properties of n-alkanes</a>, J. Math. Chemistry, 25, 1999, 55-64 (see p. 63).
%H A193530 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-7,3,-1,1,1).
%F A193530 a(n) = 1 + A005409(floor((n+3)/2)) + A107769(n).
%F A193530 From _G. C. Greubel_, May 21 2021: (Start)
%F A193530 a(n) = (1 + A001333(n) + A135153(n+2))/4.
%F A193530 a(n) = (2 + Q(n) + 2*(1+(-1)^n)*Pell((n+2)/2) + 2*(1-(-1)^n)*Pell((n+1)/2))/8.
%F A193530 a(2*n) = (2 + Q(2*n) + 4*Pell(n+1))/8.
%F A193530 a(2*n+1) = (2 + Q(2*n+1) + 4*Pell(n+1))/8, where Pell(n) = A000129(n), and Q(n) = A002203. (End)
%p A193530 f:=n->if n mod 2 = 0 then (1/4)*(A001333(n-2)+A001333((n-2)/2)+A001333((n-4)/2)+1) else (1/4)*(A001333(n-2)+A001333((n-1)/2)+A001333((n-3)/2)+1); fi; # produces the sequence with a different offset
%t A193530 LinearRecurrence[{3,1,-7,3,-1,1,1}, {1,1,2,3,7,13,31}, 40] (* _Vincenzo Librandi_, Aug 28 2016 *)
%t A193530 Table[(2 +LucasL[n, 2] +2*(1+(-1)^n)*Fibonacci[(n+2)/2, 2] + 2*(1-(-1)^n)*Fibonacci[(n+1)/2, 2])/8, {n, 0, 40}] (* _G. C. Greubel_, May 21 2021 *)
%o A193530 (Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-2*x-2*x^2 +3*x^3+x^5)/((1-x)*(1-2*x-x^2)*(1-2*x^2-x^4)) )); // _Vincenzo Librandi_, Aug 28 2016
%o A193530 (Sage)
%o A193530 @CachedFunction
%o A193530 def Pell(n): return n if (n<2) else 2*Pell(n-1) + Pell(n-2)
%o A193530 def A193530(n): return (1 + Pell(n+1) - Pell(n) + (1 + (-1)^n)*Pell((n+2)/2) + (1-(-1)^n)*Pell((n+1)/2) )/4
%o A193530 [A193530(n) for n in (0..40)] # _G. C. Greubel_, May 21 2021
%Y A193530 Cf. A000129, A001333, A002203, A135153.
%K A193530 nonn,easy
%O A193530 0,3
%A A193530 _F. Chapoton_ and _N. J. A. Sloane_, Jul 29 2011