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 A077996 #30 Aug 15 2022 10:27:16
%S A077996 1,1,3,9,23,61,163,433,1151,3061,8139,21641,57543,153005,406835,
%T A077996 1081761,2876367,7648165,20336219,54073337,143779223,382304221,
%U A077996 1016534339,2702931345,7187005471,19110010965,50812890091,135109802089,359252516199,955240614669
%N A077996 Expansion of (1-x)/(1-2*x-x^2-2*x^3).
%C A077996 From _Greg Dresden_ and _Hanzhang Fang_, Aug 12 2022: (Start)
%C A077996 a(n) is also the number of ways to tile a long box of dimensions 2 X 2 X n with "plates" (of dimension 2 X 2 X 1) and "hinges" (which are 2 X 2 X 1 plates with a 2 X 1 X 1 box added on top or bottom of the plate). The plates and hinges are shown here:
%C A077996 ______ ______
%C A077996 / /| and / /|
%C A077996 /_____/ / /_____/ |
%C A077996 |_____|/ |___ | /
%C A077996 |_|/
%C A077996 (End)
%H A077996 G. C. Greubel, <a href="/A077996/b077996.txt">Table of n, a(n) for n = 0..1000</a>
%H A077996 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,2).
%F A077996 If p[1]=1, p[2]=2, p[i]=4, (i>2), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. - _Milan Janjic_, May 02 2010
%F A077996 a(n) = Sum_{k=1..n} Sum_{i=k..n} Sum_{j=0..k} binomial(j,-3*k+2*j+i) * 2^(k-j)*binomial(k,j)*binomial(n+k-i-1,k-1). - _Vladimir Kruchinin_, May 05 2011
%t A077996 LinearRecurrence[{2,1,2}, {1,1,3}, 40] (* or *) CoefficientList[Series[(1 -x)/(1-2*x-x^2-2*x^3), {x,0,40}], x] (* _G. C. Greubel_, Jun 27 2019 *)
%o A077996 (Maxima)
%o A077996 a(n):=sum(sum((sum(binomial(j,-3*k+2*j+i)*2^(k-j)*binomial(k,j),j,0,k) )*binomial(n+k-i-1,k-1),i,k,n),k,1,n); /* _Vladimir Kruchinin_, May 05 2011 */
%o A077996 (PARI) Vec((1-x)/(1-2*x-x^2-2*x^3)+O(x^40)) \\ _Charles R Greathouse IV_, Sep 27 2012
%o A077996 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1-2*x-x^2-2*x^3) )); // _G. C. Greubel_, Jun 27 2019
%o A077996 (Sage) ((1-x)/(1-2*x-x^2-2*x^3)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 27 2019
%o A077996 (GAP) a:=[1,1,3];; for n in [4..30] do a[n]:=2*a[n-1]+a[n-2]+2*a[n-3]; od; a; # _G. C. Greubel_, Jun 27 2019
%K A077996 nonn,easy
%O A077996 0,3
%A A077996 _N. J. A. Sloane_, Nov 17 2002