A052991 Expansion of (1-x-x^2)/(1-3x-x^2).

1, 2, 6, 20, 66, 218, 720, 2378, 7854, 25940, 85674, 282962, 934560, 3086642, 10194486, 33670100, 111204786, 367284458, 1213058160, 4006458938, 13232434974, 43703763860, 144343726554, 476734943522, 1574548557120, 5200380614882, 17175690401766, 56727451820180

Offset: 0

Author

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

Keywords

• easy
• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1066
• Index entries for linear recurrences with constant coefficients, signature (3,1)

Crossrefs

Cf. A006190, A155161.

Programs

Maple

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

Mathematica

CoefficientList[Series[(1-x-x^2)/(1-3x-x^2),{x,0,30}],x] (* or *) LinearRecurrence[{3,1},{1,2,6},30] (* Harvey P. Dale, May 10 2022 *)

Formula

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

Recurrence: {a(0)=1, a(1)=2, a(n)+3*a(n+1)-a(n+2), a(2)=6}.

Sum(-2/13*(3*_alpha-2)*_alpha^(-1-n), _alpha=RootOf(-1+3*_Z+_Z^2)).

a(n) = Sum_{k=0..n} A155161(n,k)*2^k. - Philippe Deléham, Feb 08 2012

G.f.: 1/Q(0), where Q(k) = 1 + x^2 - (2*k+1)*x + x*(2*k-1 - x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 05 2013

a(n) = A006190(n+1)-A006190(n)-A006190(n-1). - R. J. Mathar, Feb 27 2019

a(n) = 2*A006190(n) for n>=1. - Philippe Deléham, Mar 09 2023

Extensions

More terms from James Sellers, Jun 06 2000