A189315 Expansion of 5*(1-3*x+x^2)/(1-5*x+5*x^2).

5, 10, 30, 100, 350, 1250, 4500, 16250, 58750, 212500, 768750, 2781250, 10062500, 36406250, 131718750, 476562500, 1724218750, 6238281250, 22570312500, 81660156250, 295449218750, 1068945312500, 3867480468750, 13992675781250, 50625976562500

Offset

0,1

Comments

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

A=A_(10,1)=

(0 1 0 0 0)

(1 0 1 0 0)

(0 1 0 1 0)

(0 0 1 0 1)

(0 0 0 2 0).

Then a(n) = Trace(A^(2*n)).

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).

From Tom Copeland, Dec 08 2015: (Start)

These are also the non-vanishing traces for the adjacency matrices of the simple Lie algebras B_5 and C_5. See links for B_4, A265185, and B_3, A025192.

a(n+1) = 10 * A081567(n), and, ignoring a(0), a G.F. is 10 *(1-2*x)/(1-5*x+5*x^2) whose denominator is y^5 * A127672(5,1/y) with y = sqrt(x).

-log(1 - 5x^2 + 5x^4) = 10 x^2/2 + 30 x^4/4 + ... provides a logarithmic series for the traces of both the odd and even powers of the matrix beginning with the first power. (End)

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

L. E. Jeffery, Unit-primitive matrices.

Index entries for linear recurrences with constant coefficients, signature (5, -5).

Formula

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

a(n) = 5*a(n-1)-5*a(n-2), n>2, a(0)=5, a(1)=10, a(2)=30.

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

a(n) = 2^(1-n)*((5-Sqrt(5))^n+(5+Sqrt(5))^n), for n>0, with a(0)=5.

a(n) = 5*A147748(n).

Mathematica

CoefficientList[Series[5(1-3x+x^2)/(1-5x+5x^2), {x, 0, 40}], x] (* or *)
Join[{5}, LinearRecurrence[{5, -5}, {10, 30}, 40]]  (* Harvey P. Dale, Apr 25 2011 *)

Prog

(PARI) Vec(5*(1-3*x+x^2)/(1-5*x+5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012
(Magma) I:=[5, 10, 30]; [n le 3 select I[n] else 5*Self(n-1)-5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 09 2015

Crossrefs

Cf. A147748, A189316, A189317, A189318.

Cf. A025192, A081567, A127672, A265185.

Sequence in context: A002571 A077916 A358728 * A056422 A032296 A052648

Adjacent sequences: A189312 A189313 A189314 * A189316 A189317 A189318

Keyword

nonn,easy

Author

L. Edson Jeffery, Apr 20 2011

Status

approved