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A189316 Expansion of 5*(1-x-x^2)/((1+x)*(1-3*x+x^2)) 5
5, 5, 15, 35, 95, 245, 645, 1685, 4415, 11555, 30255, 79205, 207365, 542885, 1421295, 3720995, 9741695, 25504085, 66770565, 174807605, 457652255, 1198149155, 3136795215, 8212236485, 21499914245, 56287506245, 147362604495, 385800307235, 1010038317215 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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

A=A_(10,2)=

(0 0 1 0 0)

(0 1 0 1 0)

(1 0 1 0 1)

(0 1 0 2 0)

(0 0 2 0 1).

Then a(n)=Trace(A^n). For m=1,2,..., A^(m) can also be written

A^(m)=

[   F(m-1)^2     0     F(m)^2      0      F(m-1)*F(m)      ]

[       0     F(2*m-1)    0      F(2*m)        0           ]

[    F(m)^2      0    F(m+1)^2     0      F(m)*F(m+1)      ]

[       0      F(2*m)     0     F(2*m+1)       0           ]

[ 2*F(m-1)*F(m)  0  2*F(m)*F(m+1)  0  F(2*m+1)-F(m)*F(m+1) ],

where F(m-1)=A000045(n) are the Fibonacci numbers and m=n+1. Hence also a(n+1)=Trace(A^(n+1))=F(m-1)^2+F(2*m-1)+F(m+1)^2+2*F(2*m+1)-F(m)*F(m+1). (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..28.

L. E. Jeffery, Unit-primitive matrices.

Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

FORMULA

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

a(n)=2*a(n-1)+2*a(n-2)-a(n-3), n>2, a(0)=5, a(1)=5, a(2)=15.

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

a(n)=(-1)^n+2*(1/tau^(2*n)+tau^(2*n)), tau=(1+Sqrt(5))/2=1.618033....

a(n)=5*A061646(n), n>=0 (offset for A061646 is -1).

MATHEMATICA

CoefficientList[Series[5 (1-x-x^2)/((1+x)(1-3x+x^2)), {x, 0, 40}], x] (* or *) LinearRecurrence[{2, 2, -1}, {5, 5, 15}, 40] (* Harvey P. Dale, Nov 26 2016 *)

CROSSREFS

Cf. A000045, A061646, A189315, A189317, A189318.

Sequence in context: A178821 A145599 A275016 * A154232 A194615 A195465

Adjacent sequences:  A189313 A189314 A189315 * A189317 A189318 A189319

KEYWORD

nonn

AUTHOR

L. Edson Jeffery, Apr 20 2011

STATUS

approved

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Last modified June 20 19:36 EDT 2019. Contains 324234 sequences. (Running on oeis4.)