OFFSET
0,3
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).
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
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 *)
PROG
(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
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1-2*x-x^2-2*x^3) )); // G. C. Greubel, Jun 27 2019
(Sage) ((1-x)/(1-2*x-x^2-2*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jun 27 2019
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Nov 17 2002
STATUS
approved