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 A167993 #28 Nov 05 2017 16:21:11
%S A167993 0,0,1,3,12,36,117,351,1080,3240,9801,29403,88452,265356,796797,
%T A167993 2390391,7173360,21520080,64566801,193700403,581120892,1743362676,
%U A167993 5230147077,15690441231,47071500840,141214502520,423644039001,1270932117003,3812797945332,11438393835996
%N A167993 Expansion of x^2/((3*x-1)*(3*x^2-1)).
%C A167993 The terms satisfy a(n) = 3*a(n-1) +3*a(n-2) -9*a(n-3), so they follow the pattern a(n) = p*a(n-1) +q*a(n-2) -p*q*a(n-3) with p=q=3. This could be called the principal sequence for that recurrence because we have set all but one of the initial terms to zero. [p=q=1 leads to the principal sequence A004526. p=q=2 leads essentially to A032085. The common feature is that the denominator of the generating function does not have a root at x=1, so the sequences of higher order successive differences have the same recurrence as the original sequence. See A135094, A010036, A006516.]
%H A167993 Vincenzo Librandi, <a href="/A167993/b167993.txt">Table of n, a(n) for n = 0..200</a>
%H A167993 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,-9).
%F A167993 a(2*n+1) = 3*a(2*n).
%F A167993 a(2*n) = A122006(2*n)/2.
%F A167993 a(n) = 3*a(n-1) + 3*a(n-4) - 9*a(n-3).
%F A167993 a(n+1) - a(n) = A122006(n).
%F A167993 a(n) = (3^n - A108411(n+1))/6.
%F A167993 G.f.: x^2/((3*x-1)*(3*x^2-1)).
%F A167993 From _Colin Barker_, Sep 23 2016: (Start)
%F A167993 a(n) = 3^(n-1)/2-3^(n/2-1)/2 for n even.
%F A167993 a(n) = 3^(n-1)/2-3^(n/2-1/2)/2 for n odd.
%F A167993 (End)
%t A167993 CoefficientList[Series[x^2/((3*x - 1)*(3*x^2 - 1)), {x, 0, 50}], x] (* _G. C. Greubel_, Jul 03 2016 *)
%t A167993 LinearRecurrence[{3,3,-9},{0,0,1},30] (* _Harvey P. Dale_, Nov 05 2017 *)
%o A167993 (PARI) Vec(x^2/((3*x-1)*(3*x^2-1))+O(x^99)) \\ _Charles R Greathouse IV_, Jun 29 2011
%Y A167993 Cf. A138587, A107767 (partial sums).
%K A167993 nonn,easy
%O A167993 0,4
%A A167993 _Paul Curtz_, Nov 16 2009
%E A167993 Formulae corrected by _Johannes W. Meijer_, Jun 28 2011