A189316 Expansion of g.f. 5*(1-x-x^2)/((1+x)*(1-3*x+x^2)).

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

Offset: 0

Author

L. Edson Jeffery, Apr 20 2011

Keywords

• nonn
• easy

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

Links

• L. E. Jeffery, Unit-primitive matrices.
• Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Crossrefs

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

Programs

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 *)

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).

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).

E.g.f.: cosh(x) + 4*exp(3*x/2)*cosh(sqrt(5)*x/2) - sinh(x). - Stefano Spezia, Jul 09 2024

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