A052899 Expansion of g.f.: ( 1-2*x ) / ((x-1)*(4*x^2+2*x-1)).

1, 1, 5, 13, 45, 141, 461, 1485, 4813, 15565, 50381, 163021, 527565, 1707213, 5524685, 17878221, 57855181, 187223245, 605867213, 1960627405, 6344723661, 20531956941, 66442808525, 215013444813, 695798123725, 2251650026701

Offset

0,3

Comments

From L. Edson Jeffery, Apr 19 2011: (Start)

Let A be the unit-primitive matrix (see [Jeffery])

A = A_(10,4) =

(0 0 0 0 1)

(0 0 0 2 0)

(0 0 2 0 1)

(0 2 0 2 0)

(2 0 2 0 1).

Then a(n) = (1/5)*trace(A^n). (End)

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 875

L. E. Jeffery, Unit-primitive matrices

Index entries for linear recurrences with constant coefficients, signature (3,2,-4).

Formula

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

a(n) = Sum(-1/25*(-1-8*_alpha+4*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(1-3*_Z-2*_Z^2+4*_Z^3)).

a(n)/a(n-1) tends to (1 + sqrt(5)) = 3.236067... - Gary W. Adamson, Mar 01 2008

a(n) = (1/5) * Sum_{k=1..5} ((x_k)^4-3*(x_k)^2+1), x_k=2*cos((2*k-1)*Pi/10). Also, a(n)/a(n-1) -> spectral radius of matrix A_(10,4) above. - L. Edson Jeffery, Apr 19 2011

a(n) = (2*A087131(n)+1)/5. - Bruno Berselli, Apr 20 2011

Maple

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

Mathematica

CoefficientList[Series[(1-2x)/((x-1)(4x^2+2x-1)), {x, 0, 40}], x] (* or *) LinearRecurrence[{3, 2, -4}, {1, 1, 5}, 40] (* Harvey P. Dale, Jul 10 2017 *)

Prog

(Sage) from sage.combinat.sloane_functions import recur_gen2b
it = recur_gen2b(1, 1, 2, 4, lambda n:-1)
[next(it) for i in range(1, 28)] # Zerinvary Lajos, Jul 09 2008
(Magma) [(1/5)*(2^(n+1)*Lucas(n)+1): n in [0..50]]; // Vincenzo Librandi, Apr 20 2011
(Maxima)  makelist(coeff(taylor((1-2*x)/(1-3*x-2*x^2+4*x^3), x, 0, n), x, n), n, 0, 25); /* Bruno Berselli, May 30 2011 */

Crossrefs

Cf. A084057.

Sequence in context: A113835 A006349 A322203 * A147200 A147396 A320877

Adjacent sequences: A052896 A052897 A052898 * A052900 A052901 A052902

Keyword

easy,nonn

Author

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

Extensions

More terms from James A. Sellers, Jun 08 2000

Status

approved