A077997 Expansion of (1-x)/(1-2*x-x^2-x^3).

1, 1, 3, 8, 20, 51, 130, 331, 843, 2147, 5468, 13926, 35467, 90328, 230049, 585893, 1492163, 3800268, 9678592, 24649615, 62778090, 159884387, 407196479, 1037055435, 2641191736, 6726635386, 17131517943, 43630863008, 111119879345, 283002139641, 720755021635

Offset: 0

Author

N. J. A. Sloane, Nov 17 2002

Keywords

• nonn
• easy

Comments

Number of compositions of n where there is one sort of part 1, two sorts of part 2, and three sorts of every other part. - Joerg Arndt, Mar 15 2014

Number of ways to tile a strip of length n with red and blue squares, blue dominos, and blue trominos, where the first tile must be blue. - Greg Dresden and Arnim Kuchhal, Aug 05 2024

Links

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

Programs

GAP

a:=[1,1,3];; for n in [4..30] do a[n]:=2*a[n-1]+a[n-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-x^3) )); // G. C. Greubel, Jun 27 2019

Mathematica

CoefficientList[Series[(1-x)/(1-2x-x^2-x^3),{x,0,30}],x] (* or *) LinearRecurrence[{2,1,1},{1,1,3},30] (* Harvey P. Dale, May 24 2011 *)

Maxima

a(n):=sum(sum(binomial(m+i-1,m-1) * sum(binomial(j,n-3*m+2*j-i) * binomial(m,j),j,0,m), i,0,n-m), m,1,n); /* Vladimir Kruchinin, May 12 2011 */

PARI

Vec((1-x)/(1-2*x-x^2-x^3)+O(x^30)) \\ Charles R Greathouse IV, Sep 26 2012

Sage

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

Formula

a(n) = 2*a(n-1) + a(n-2) + a(n-3), n>2, with a(0)=1, a(1)=1, a(2)=3. - Philippe Deléham, Nov 20 2008

If p[1]=1, p[2]=2, p[i]=3, (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_{m=1..n} Sum_{i=0..n-m} binomial(m+i-1,m-1)*Sum_{j=0..m} binomial(j,n-3*m+2*j-i)*binomial(m,j), n>0, a(0)=1. - Vladimir Kruchinin, May 12 2011

Extensions

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021