A052953 Expansion of 2*(1-x-x^2)/((1-x)*(1+x)*(1-2*x)).

2, 2, 4, 6, 12, 22, 44, 86, 172, 342, 684, 1366, 2732, 5462, 10924, 21846, 43692, 87382, 174764, 349526, 699052, 1398102, 2796204, 5592406, 11184812, 22369622, 44739244, 89478486, 178956972, 357913942, 715827884, 1431655766, 2863311532

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

a(n) = sum of absolute values of terms in the (n+1)-th row of the triangle in A108561; - Reinhard Zumkeller, Jun 10 2005

a(n) = A078008(n+1) + 2*(1 + n mod 2). - Reinhard Zumkeller, Jun 10 2005

Essentially the same as A128209. - R. J. Mathar, Jun 14 2008

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1024
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Crossrefs

Apart from initial term, equals A026644(n+1) + 2.

Cf. A001045.

Programs

GAP

List([0..40], n-> (2^(n+1) +3 +(-1)^n)/3); # G. C. Greubel, Oct 21 2019

Magma

[(2^(n+1) +3 +(-1)^n)/3: n in [0..40]]; // G. C. Greubel, Oct 21 2019

Maple

spec:= [S,{S=Union(Sequence(Union(Prod(Union(Z,Z),Z),Z)),Sequence(Z))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq((2^(n+1) +3 +(-1)^n)/3, n=0..40); # G. C. Greubel, Oct 21 2019

Mathematica

LinearRecurrence[{2,1,-2}, {2,2,4}, 40] (* G. C. Greubel, Oct 22 2019 *)
CoefficientList[Series[2(1-x-x^2)/((1-x)(1+x)(1-2x)),{x,0,40}],x] (* Harvey P. Dale, Aug 03 2025 *)

PARI

vector(41, n, (2^n +3 -(-1)^n)/3 ) \\ G. C. Greubel, Oct 21 2019

Sage

[(2^(n+1) +3 +(-1)^n)/3 for n in (0..40)] # G. C. Greubel, Oct 21 2019

Formula

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

a(n) = a(n-1) + 2*a(n-2) - 2.

a(n) = 1 + Sum_{alpha=RootOf(-1+z+2*z^2)} (1 + 4*alpha)*alpha^(-1-n)/9.

a(2n) = 2*a(n-1)-2, a(2n+1) = 2*a(2n). - Lee Hae-hwang, Oct 11 2002

From Paul Barry, May 24 2004: (Start)

a(n) = A001045(n+1) + 1.

a(n) = (2^(n+1) - (-1)^(n+1))/3 + 1. (End)

E.g.f.: (2*exp(2*x) + 3*exp(x) + exp(-x))/3. - G. C. Greubel, Oct 21 2019

Extensions

More terms from James Sellers, Jun 05 2000