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 A052543 #48 Jul 02 2025 16:01:58
%S A052543 1,2,8,26,90,306,1046,3570,12190,41618,142094,485138,1656366,5655186,
%T A052543 19308014,65921682,225070702,768439442,2623616366,8957586578,
%U A052543 30583113582,104417281170,356502897518,1217177027730,4155702315886
%N A052543 Expansion of (1-x)/(1 - 3*x - 2*x^2 + 2*x^3).
%C A052543 From _Andrew Woods_, Jun 03 2013: (Start)
%C A052543 a(n) is the number of ways to tile a 2 X n square grid with 1 X 1, 1 X 2, 2 X 1, and 2 X 2 tiles. Solutions for a(2)=8:
%C A052543 . _ _ _ _ ___ ___ ___ _ _ _ _ _ _
%C A052543 | | | | |_| | | |___| |___| |_| | |_|_| |_|_|
%C A052543 |_|_| |_|_| |___| |___| |_|_| |_|_| |___| |_|_|
%C A052543 (End)
%H A052543 G. C. Greubel, <a href="/A052543/b052543.txt">Table of n, a(n) for n = 0..1000</a>
%H A052543 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=478">Encyclopedia of Combinatorial Structures 478</a>
%H A052543 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,2,-2).
%F A052543 G.f.: (1-x)/(1 - 3*x - 2*x^2 + 2*x^3).
%F A052543 a(n) = 3*a(n-1) + 2*a(n-2) - 2*a(n-3), with a(0)=1, a(1)=2, a(2)=8.
%F A052543 a(n) = Sum_{alpha = RootOf(1 -3*x -2*x^2 +2*x^3)} (1/98)*(13 + 25*alpha - 16*alpha^2)*alpha^(-n-1).
%F A052543 Equals triangle A059260 * the Pell sequence [1, 2, 5, 12, ...] as a vector. - _Gary W. Adamson_, Mar 06 2012
%F A052543 a(n) = A214997(n) - A214996(n). - _Clark Kimberling_, Nov 28 2012
%p A052543 spec := [S,{S=Sequence(Prod(Union(Z,Z),Union(Z,Sequence(Z))))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052543 CoefficientList[Series[(1-x)/(1-3x-2x^2+2x^3),{x,0,30}],x] (* or *) LinearRecurrence[{3,2,-2},{1,2,8},30] (* _Harvey P. Dale_, Jan 23 2013 *)
%o A052543 (PARI) my(x='x+O('x^30)); Vec((1-x)/(1-3*x-2*x^2+2*x^3)) \\ _G. C. Greubel_, May 09 2019
%o A052543 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1 -3*x-2*x^2+2*x^3) )); // _G. C. Greubel_, May 09 2019
%o A052543 (Sage) ((1-x)/(1-3*x-2*x^2+2*x^3)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 09 2019
%o A052543 (GAP) a:=[1,2,8];; for n in [4..30] do a[n]:=3*a[n-1]+2*a[n-2]-2*a[n-3]; od; a; # _G. C. Greubel_, May 09 2019
%Y A052543 Cf. A000129, A059260.
%K A052543 easy,nonn
%O A052543 0,2
%A A052543 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052543 More terms from _James Sellers_, Jun 06 2000