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 A083881 #22 Sep 08 2022 08:45:10
%S A083881 1,3,12,54,252,1188,5616,26568,125712,594864,2814912,13320288,
%T A083881 63032256,298271808,1411437312,6678993024,31605334272,149558047488,
%U A083881 707716279296,3348949390848,15847398669312,74990695670784,354859782008832
%N A083881 a(n) = 6*a(n-1) - 6*a(n-2), with a(0)=1, a(1)=3.
%C A083881 Binomial transform of A001075.
%H A083881 G. C. Greubel, <a href="/A083881/b083881.txt">Table of n, a(n) for n = 0..1000</a>
%H A083881 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-6).
%F A083881 a(n) = ((3-sqrt(3))^n + (3+sqrt(3))^n)/2.
%F A083881 a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*3^(n-k).
%F A083881 G.f.: (1-3*x)/(1-6*x+6*x^2).
%F A083881 E.g.f.: exp(3*x) * cosh(x*sqrt(3)).
%F A083881 a(n) = right and left terms in M^n * [1 1 1] where M = the 3X3 matrix [1 1 1 / 1 4 1 / 1 1 1]. M^n * [1 1 1] = [a(n) A030192(n) a(n)]. E.g. a(3) = 54 since M^3 * [1 1 1] = [54 144 54] = [a(3) A030192(3) a(3)]. - _Gary W. Adamson_, Dec 18 2004
%F A083881 a(n) = Sum_{k, 0<=k<=n}3^k*A098158(n,k). - _Philippe Deléham_, Dec 04 2006
%F A083881 G.f.: A(x) = G(0) where G(k) = 1 + 3*x/((1-3*x) - x*(1-3*x)/(x + (1-3*x)/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Dec 29 2012.
%t A083881 f[n_]:= Simplify[(3 + Sqrt@3)^n + (3 - Sqrt@3)^n]/2; Array[f, 30, 0] (* _Robert G. Wilson v_, Oct 31 2010 *)
%t A083881 LinearRecurrence[{6,-6}, {1,3}, 30] (* _G. C. Greubel_, Aug 01 2019 *)
%o A083881 (PARI) my(x='x+O('x^30)); Vec((1-3*x)/(1-6*x+6*x^2)) \\ _G. C. Greubel_, Aug 01 2019
%o A083881 (Magma) I:=[1,3]; [n le 2 select I[n] else 6*Self(n-1) -6*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Aug 01 2019
%o A083881 (Sage) ((1-3*x)/(1-6*x+6*x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 01 2019
%o A083881 (GAP) a:=[1,3];; for n in [3..30] do a[n]:=6*a[n-1]-6*a[n-2]; od; a; # _G. C. Greubel_, Aug 01 2019
%Y A083881 Cf. A083882.
%Y A083881 Cf. A030192.
%K A083881 easy,nonn
%O A083881 0,2
%A A083881 _Paul Barry_, May 08 2003