A052971 Expansion of (1-x)/(1-2*x-2*x^3+2*x^4).

1, 1, 2, 6, 12, 26, 60, 132, 292, 652, 1448, 3216, 7152, 15896, 35328, 78528, 174544, 387952, 862304, 1916640, 4260096, 9468896, 21046464, 46779840, 103977280, 231109696, 513686144, 1141767168, 2537799168, 5640751232, 12537664512, 27867393024, 61940690176

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

a(n) is the number of compositions of n using three colors of 3. Compare to A077998 which gives the number of compositions of n using two colors of 2. - Greg Dresden and Yushu Fan, Aug 15 2023

Links

• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1043
• Index entries for linear recurrences with constant coefficients, signature (2,0,2,-2).

Crossrefs

Cf. A077998.

Programs

Maple

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

Mathematica

CoefficientList[Series[(1-x)/(1-2x-2x^3+2x^4),{x,0,30}],x] (* or *) LinearRecurrence[{2,0,2,-2},{1,1,2,6},32] (* Harvey P. Dale, Jul 23 2012 *)

Formula

G.f.: -(-1+x)/(1-2*x-2*x^3+2*x^4).

Recurrence: {a(1)=1, a(0)=1, a(3)=6, a(2)=2, 2*a(n)-2*a(n+1)-2*a(n+3)+a(n+4)=0}.

Sum(-1/227*(-29-50*_alpha+45*_alpha^3-14*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(1-2*_Z-2*_Z^3+2*_Z^4)).

Extensions

More terms from James Sellers, Jun 06 2000