A052543 Expansion of (1-x)/(1 - 3*x - 2*x^2 + 2*x^3).

1, 2, 8, 26, 90, 306, 1046, 3570, 12190, 41618, 142094, 485138, 1656366, 5655186, 19308014, 65921682, 225070702, 768439442, 2623616366, 8957586578, 30583113582, 104417281170, 356502897518, 1217177027730, 4155702315886

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

From Andrew Woods, Jun 03 2013: (Start)

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:

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(End)

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 478
• Index entries for linear recurrences with constant coefficients, signature (3,2,-2).

Crossrefs

Cf. A000129, A059260.

Programs

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

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1 -3*x-2*x^2+2*x^3) )); // G. C. Greubel, May 09 2019

Maple

spec := [S,{S=Sequence(Prod(Union(Z,Z),Union(Z,Sequence(Z))))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

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 *)

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

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

Formula

G.f.: (1-x)/(1 - 3*x - 2*x^2 + 2*x^3).

a(n) = 3*a(n-1) + 2*a(n-2) - 2*a(n-3), with a(0)=1, a(1)=2, a(2)=8.

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).

Equals triangle A059260 * the Pell sequence [1, 2, 5, 12, ...] as a vector. - Gary W. Adamson, Mar 06 2012

a(n) = A214997(n) - A214996(n). - Clark Kimberling, Nov 28 2012

Extensions

More terms from James Sellers, Jun 06 2000