A105058 Expansion of g.f. (1+8*x-x^2)/((1+x)*(1-6*x+x^2)).

1, 13, 69, 409, 2377, 13861, 80781, 470833, 2744209, 15994429, 93222357, 543339721, 3166815961, 18457556053, 107578520349, 627013566049, 3654502875937, 21300003689581, 124145519261541, 723573111879673

Offset

0,2

Comments

A floretion-generated sequence relating the squares of the numerators of continued fraction convergents to sqrt(2) to the squares of the denominators of continued fraction convergents to sqrt(2) (Pell numbers).

Floretion Algebra Multiplication Program, FAMP Code:

1dia[J]tesseq[ - .5'j + .5'k - .5j' + .5k' - 2'ii' + 'jj' - 'kk' + .5'ij' + .5'ik' + .5'ji' + 'jk' + .5'ki' + 'kj' + e ]. Identity used: dia[I]tes + dia[J]tes + dia[K]tes = jes + fam + 3tes.

Links

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

Index entries for linear recurrences with constant coefficients, signature (5,5,-1)

Formula

a(n) = 2 * A001109(n+1) - (-1)^n.

G.f.: G(0)/(1-3*x) - 1/(1+x), where G(k) = 1 + 1/(1 - x*(8*k-9)/( x*(8*k-1) - 3/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 12 2013

From G. C. Greubel, Aug 21 2022: (Start)

a(n) = A000129(2*n+2) - (-1)^n.

E.g.f.: exp(3*x)*( 2*cosh(2*sqrt(2)*x) + (3/sqrt(2))*sinh(2*sqrt(2)*x)) - exp(-x). (End)

Mathematica

CoefficientList[ Series[(1+8x-x^2)/((1+x)(1-6x+x^2)), {x, 0, 30}], x] (* Robert G. Wilson v, Apr 06 2005 *)
LinearRecurrence[{5, 5, -1}, {1, 13, 69}, 30] (* Harvey P. Dale, Jun 03 2017 *)

Prog

(Magma) [Evaluate(DicksonSecond(2*n+1, -1), 2) -(-1)^n: n in [0..30]]; // G. C. Greubel, Aug 21 2022
(SageMath) [lucas_number1(2*n+2, 2, -1) -(-1)^n for n in (0..30)] # G. C. Greubel, Aug 21 2022

Crossrefs

Cf. A000129, A001109, A001333, A046729, A077444.

Cf. A077445, A079291, A082639, A084158, A090390.

Sequence in context: A137188 A055338 A055880 * A146469 A146381 A085461

Adjacent sequences: A105055 A105056 A105057 * A105059 A105060 A105061

Keyword

nonn,easy

Author

Creighton Dement, Apr 04 2005

Status

approved