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

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

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 December 14 01:27 EST 2019. Contains 329978 sequences. (Running on oeis4.)