

A189315


Expansion of 5*(13*x+x^2)/(15*x+5*x^2).


8



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
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OFFSET

0,1


COMMENTS

Let A be the unitprimitive 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 unitprimitive matrix A_(N,r) (0<r<floor(N/2)) and for which the closedform expression for a(n) is derived from the eigenvalues of A_(N,r).
These are also the nonvanishing 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 *(12*x)/(15*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



FORMULA

G.f.: 5*(13*x+x^2)/(15*x+5*x^2).
a(n) = 5*a(n1)5*a(n2), 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*k1)*Pi/10).
a(n) = 2^(1n)*((5Sqrt(5))^n+(5+Sqrt(5))^n), for n>0, with a(0)=5.


MATHEMATICA

CoefficientList[Series[5(13x+x^2)/(15x+5x^2), {x, 0, 40}], x] (* or *)
Join[{5}, LinearRecurrence[{5, 5}, {10, 30}, 40]] (* Harvey P. Dale, Apr 25 2011 *)


PROG

(Magma) I:=[5, 10, 30]; [n le 3 select I[n] else 5*Self(n1)5*Self(n2): n in [1..30]]; // Vincenzo Librandi, Dec 09 2015


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



