A077996 Expansion of (1-x)/(1-2*x-x^2-2*x^3).

1, 1, 3, 9, 23, 61, 163, 433, 1151, 3061, 8139, 21641, 57543, 153005, 406835, 1081761, 2876367, 7648165, 20336219, 54073337, 143779223, 382304221, 1016534339, 2702931345, 7187005471, 19110010965, 50812890091, 135109802089, 359252516199, 955240614669

Offset: 0

Author

N. J. A. Sloane, Nov 17 2002

Keywords

• nonn
• easy

Comments

From Greg Dresden and Hanzhang Fang, Aug 12 2022: (Start)

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:

____ ____

/ /| and / /|

/___/ / /___/ |

|___|/ |_ | /

|_|/

(End)

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1,2).

Programs

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

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1-2*x-x^2-2*x^3) )); // G. C. Greubel, Jun 27 2019

Mathematica

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 *)

Maxima

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 */

PARI

Vec((1-x)/(1-2*x-x^2-2*x^3)+O(x^40)) \\ Charles R Greathouse IV, Sep 27 2012

Sage

((1-x)/(1-2*x-x^2-2*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jun 27 2019

Formula

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

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