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A052953 Expansion of 2*(1-x-x^2)/((1-x)*(1+x)*(1-2*x)). 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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).

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

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 *)

PROG

(Sage) from sage.combinat.sloane_functions import recur_gen2; it = recur_gen2(1, 1, 1, 2); [it.next()+1 for i in xrange(0, 34)] # Zerinvary Lajos, Jul 06 2008

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

(PARI) vector(41, n, (2^n +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

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

CROSSREFS

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

Cf. A001045.

Sequence in context: A216957 A122536 A238014 * A128209 A274935 A188538

Adjacent sequences:  A052950 A052951 A052952 * A052954 A052955 A052956

KEYWORD

easy,nonn,changed

AUTHOR

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

EXTENSIONS

More terms from James A. Sellers, Jun 05 2000

STATUS

approved

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Last modified October 23 07:05 EDT 2019. Contains 328335 sequences. (Running on oeis4.)