A052973 Expansion of ( 1-x ) / ( 1-x-3*x^2+x^3 ).

1, 0, 3, 2, 11, 14, 45, 76, 197, 380, 895, 1838, 4143, 8762, 19353, 41496, 90793, 195928, 426811, 923802, 2008307, 4352902, 9454021, 20504420, 44513581, 96572820, 209609143, 454814022, 987068631, 2141901554, 4648293425

Offset: 0

Author

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

Keywords

• easy
• nonn

References

• Kenneth Edwards, Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169-177.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1045
• Index entries for linear recurrences with constant coefficients, signature (1,3,-1).

Programs

Maple

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

Mathematica

CoefficientList[Series[(1-x)/(1-x-3x^2+x^3),{x,0,30}],x] (* or *) LinearRecurrence[{1,3,-1},{1,0,3},40] (* Harvey P. Dale, Sep 06 2017 *)

Formula

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

Recurrence: {a(1)=0, a(0)=1, a(2)=3, a(n)-3*a(n+1)-a(n+2)+a(n+3)=0}

Sum(-1/74*(1-34*_alpha+9*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(1-_Z-3*_Z^2+_Z^3))

a(n) = A125691(n)-A125691(n-1). - R. J. Mathar, Feb 27 2019

Extensions

More terms from James Sellers, Jun 06 2000