A052951 Expansion of (1 + x - 2*x^2)/(1 - 2*x)^2.

1, 5, 14, 36, 88, 208, 480, 1088, 2432, 5376, 11776, 25600, 55296, 118784, 253952, 540672, 1146880, 2424832, 5111808, 10747904, 22544384, 47185920, 98566144, 205520896, 427819008, 889192448, 1845493760, 3825205248, 7918845952

Offset

0,2

Comments

Equals binomial transform of A042948 starting with "1": (1, 4, 5, 8, 9, 12, 13, ...) = terms > 0, == 0 or 1 mod 4. - Gary W. Adamson, Feb 07 2009

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

O. Aichholzer, A. Asinowski, and T. Miltzow, Disjoint compatibility graph of non-crossing matchings of points in convex position, arXiv preprint arXiv:1403.5546 [math.CO], 2014.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1021.

Agustín Moreno Cañadas, Hernán Giraldo, Gabriel Bravo Rios, On the Number of Sections in the Auslander-Reiten Quiver of Algebras of Dynkin Type, Far East Journal of Mathematical Sciences (FJMS), Vol. 101, No. 8 (2017), pp. 1631-1654.

Index entries for linear recurrences with constant coefficients, signature (4,-4).

Formula

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

a(n) = 4*(a(n-1) - a(n-2)).

a(n) = (n+1)*2^n + 2^(n-1), n > 0.

a(n) = A118413(n+1,n-1) for n > 2. - Reinhard Zumkeller, Apr 27 2006

E.g.f.: (1/2)*(-1 + exp(2*x)*(3 + 4*x)). - Stefano Spezia, Oct 22 2019

From Amiram Eldar, Oct 01 2022: (Start)

Sum_{n>=0} 1/a(n) = 4*sqrt(2)*arcsinh(1) - 11/3.

Sum_{n>=0} (-1)^n/a(n) = 13/3 - 4*sqrt(2)*arccot(sqrt(2)). (End)

Maple

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

Mathematica

CoefficientList[Series[(1+x-2*x^2)/(1-2*x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 22 2012 *)
LinearRecurrence[{4, -4}, {1, 5, 14}, 40] (* G. C. Greubel, Oct 21 2019 *)

Prog

(Magma) I:=[1, 5, 14]; [n le 3 select I[n] else 4*Self(n-1)-4*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Jun 22 2012
(PARI) x='x+O('x^40); Vec((1+x-2*x^2)/(1-2*x)^2) \\ Altug Alkan, Mar 03 2018
(Sage) [1]+[2^(n-1)*(2*n+3) for n in (1..40)] # G. C. Greubel, Oct 21 2019
(GAP) Concatenation([1], List([1..40], n-> 2^(n-1)*(2*n+3) )); # G. C. Greubel, Oct 21 2019

Crossrefs

Cf. A042948, A118413.

Sequence in context: A193557 A187198 A097507 * A048745 A307462 A292170

Adjacent sequences: A052948 A052949 A052950 * A052952 A052953 A052954

Keyword

easy,nonn

Author

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

Status

approved