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 A081340 #29 Dec 07 2019 12:18:24
%S A081340 1,2,13,62,313,1562,7813,39062,195313,976562,4882813,24414062,
%T A081340 122070313,610351562,3051757813,15258789062,76293945313,381469726562,
%U A081340 1907348632813,9536743164062,47683715820313,238418579101562
%N A081340 (5^n+(-1)^n)/2.
%C A081340 Binomial transform of A003665. 2nd binomial transform of (1,0,9,0,81,0,729,0,..). Case k=2 of family of recurrences a(n)=2k*a(n-1)-(k^2-9)*a(n-2), a(0)=0, a(1)=k. A003665 is case k=1.
%H A081340 Vincenzo Librandi, <a href="/A081340/b081340.txt">Table of n, a(n) for n = 0..200</a>
%H A081340 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,5).
%F A081340 a(n) = 4*a(n-1) + 5*a(n-2), a(0)=1, a(1)=2.
%F A081340 G.f.: (1-2*x)/((1+x)*(1-5*x)).
%F A081340 E.g.f.: exp(2*x) * cosh(3*x).
%F A081340 a(n) = ((2+sqrt(9))^n+(2-sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008
%F A081340 a(n) = sum( k=0..n, A201730(n,k)*8^k ). - _Philippe Deléham_, Dec 06 2011
%t A081340 CoefficientList[Series[(1 - 2 x) / ((1 + x) (1 - 5 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Aug 08 2013 *)
%o A081340 (Sage) [lucas_number2(n,4,-5)/2 for n in range(0, 22)] # _Zerinvary Lajos_, May 14 2009
%o A081340 (PARI) a(n)=(5^n+(-1)^n)/2 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A081340 Cf. A081341, A087404.
%K A081340 nonn,easy
%O A081340 0,2
%A A081340 _Paul Barry_, Mar 18 2003