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A052963 a(0)=1, a(1)=2, a(2)=5, a(n) = 3*a(n+2) - a(n+3). 3
1, 2, 5, 14, 40, 115, 331, 953, 2744, 7901, 22750, 65506, 188617, 543101, 1563797, 4502774, 12965221, 37331866, 107492824, 309513251, 891207887, 2566130837, 7388879260, 21275429893, 61260158842, 176391597266, 507899361905 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Equals the INVERT transform of the Pell sequence prefaced with a "1": (1, 1, 2, 5, 12, 29,...). [From Gary W. Adamson, May 01 2009]

LINKS

Table of n, a(n) for n=0..26.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1034

Sergey Kitaev, Jeffrey Remmel, Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (arXiv:1302.2274)

Index entries for linear recurrences with constant coefficients, signature (3,0,-1).

FORMULA

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

Sum(1/9*(1+2*_alpha+_alpha^2)*_alpha^(-1-n), _alpha=RootOf(1-3*_Z+_Z^3))

a(n)/a(n-1) tends to 2.8793852... = 2Cos(4)Pi/9, a root of x^3 -3x^2 + 1 (the characteristic polynomial of the 3 X 3 matrix). The latter polynomial is a factor (with (x + 1)) of the 4th degree polynomial of A066170: x^4 - 2x^3 - 3x^2 + x + 1. Given the 3 X 3 matrix [0 1 0 / 0 0 1 / -1 0 3], (M^n)*[1 1 1] = [a(n-2), a(n-1), a(n)]. - Gary W. Adamson, Feb 29 2004

a(n) = A076264(n)-A076264(n-1)-A076264(n-2). - R. J. Mathar, Feb 27 2019

MAPLE

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

MATHEMATICA

LinearRecurrence[{3, 0, -1}, {1, 2, 5}, 30] (* Harvey P. Dale, Dec 26 2015 *)

CROSSREFS

Cf. A066170.

Sequence in context: A059505 A159035 A117189 * A329275 A036908 A293346

Adjacent sequences:  A052960 A052961 A052962 * A052964 A052965 A052966

KEYWORD

easy,nonn

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 July 13 16:22 EDT 2020. Contains 335688 sequences. (Running on oeis4.)