OFFSET
0,1
COMMENTS
(Start) Let A be the unit-primitive matrix (see [Jeffery])
A=A_(10,3)=
(0 0 0 1 0)
(0 0 1 0 1)
(0 1 0 2 0)
(1 0 2 0 1)
(0 2 0 2 0).
Then a(n)=Trace(A^(2*n)). (End)
Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers (here they are A^(2*n)) 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
L. E. Jeffery, Unit-primitive matrices.
Index entries for linear recurrences with constant coefficients, signature (10, -5).
FORMULA
G.f.: 5*(1-6*x+x^2)/(1-10*x+5*x^2).
a(n)=10*a(n-1)-5*a(n-2), n>2, a(0)=5, a(1)=20, a(2)=180.
a(n)=Sum_{k=1..5} ((w_k)^3-2*w_k)^(2*n), w_k=2*cos((2*k-1)*Pi/10).
a(n)=2*((5-2*Sqrt(5))^n+(5+2*Sqrt(5))^n), for n>0, with a(0)=5.
MATHEMATICA
CoefficientList[Series[5*(1-6x+x^2)/(1-10x+5x^2), {x, 0, 30}], x] (* or *) Join[ {5}, LinearRecurrence[{10, -5}, {20, 180}, 30]] (* Harvey P. Dale, Apr 02 2013 *)
PROG
(PARI) Vec(5*(1-6*x+x^2)/(1-10*x+5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
L. Edson Jeffery, Apr 20 2011
STATUS
approved