login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A189318 Expansion of 5*(1-2*x)/(1-3*x-2*x^2+4*x^3) 5
5, 5, 25, 65, 225, 705, 2305, 7425, 24065, 77825, 251905, 815105, 2637825, 8536065, 27623425, 89391105, 289275905, 936116225, 3029336065, 9803137025, 31723618305, 102659784705, 332214042625, 1075067224065, 3478990618625, 11258250133505 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

(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)=Trace(A^n). (End)

Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers of a unit-primitive matrix A_(N,r) (0<r<floor(N/2)) and for which the closed-form expression for a(n) is derived from the eigenvalues of A_(N,r).

LINKS

Table of n, a(n) for n=0..25.

L. E. Jeffery, Unit-primitive matrices.

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

FORMULA

G.f.: 5*(1-2*x)/(1-3*x-2*x^2+4*x^3).

a(n)=3*a(n-1)+2*a(n-2)-4*a(n-3), n>3, a(0)=5, a(1)=5, a(2)=25, a(3)=65.

a(n)=Sum_{k=1..5} ((w_k)^4-3*(w_k)^2+1)^n, w_k=2*cos((2*k-1)*Pi/10).

a(n)=1+2*(1-Sqrt(5))^n+2*(1+Sqrt(5))^n.

a(n)=5*A052899(n).

MATHEMATICA

CoefficientList[Series[5(1-2x)/(1-3x-2x^2+4x^3), {x, 0, 30}], x] (* or *) LinearRecurrence[{3, 2, -4}, {5, 5, 25}, 30] (* Harvey P. Dale, Jun 02 2014 *)

PROG

(PARI) Vec(5*(1-2*x)/(1-3*x-2*x^2+4*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

CROSSREFS

Cf. A052899.

A189315, A189316, A189317.

Sequence in context: A257607 A093643 A223263 * A257615 A257624 A176160

Adjacent sequences:  A189315 A189316 A189317 * A189319 A189320 A189321

KEYWORD

nonn,easy

AUTHOR

L. Edson Jeffery, Apr 20 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 26 04:11 EDT 2019. Contains 324369 sequences. (Running on oeis4.)