A189334 Expansion of g.f. (1-6*x+x^2)/(1-10*x+5*x^2).

1, 4, 36, 340, 3220, 30500, 288900, 2736500, 25920500, 245522500, 2325622500, 22028612500, 208658012500, 1976437062500, 18721080562500, 177328620312500, 1679680800312500, 15910164901562500, 150703245014062500, 1427481625632812500, 13521300031257812500, 128075592184414062500

Offset

0,2

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)=(1/5)*Trace(A^(2*n)). (See also A189317.) (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

Table of n, a(n) for n=0..21.

L. E. Jeffery, Unit-primitive matrices.

Index entries for linear recurrences with constant coefficients, signature (10,-5).

Formula

a(n) = 10*a(n-1) - 5*a(n-2), n>2, a(0)=1, a(1)=4, a(2)=36.

a(n) = (1/5)*Sum_{k=1..5} ((w_k)^3-2*w_k)^(2*n), w_k = 2*cos((2*k-1)*Pi/10).

From Stefano Spezia, Jul 09 2024: (Start)

a(n) = 2*((5 - 2*sqrt(5))^n + (5 + 2*sqrt(5))^n)/5 for n > 0.

E.g.f.: (1 + 4*exp(5*x)*cosh(2*sqrt(5)*x))/5. (End)

Mathematica

LinearRecurrence[{10, -5}, {1, 4, 36}, 22] (* Stefano Spezia, Jul 09 2024 *)

Crossrefs

Cf. A189315, A189316, A189317, A189318.

Sequence in context: A270473 A374388 A288269 * A026334 A247562 A372465

Adjacent sequences: A189331 A189332 A189333 * A189335 A189336 A189337

Keyword

nonn,easy

Author

L. Edson Jeffery, Apr 20 2011

Extensions

a(20)-a(21) from Stefano Spezia, Jul 09 2024

Status

approved