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

Author

Creighton Dement, Apr 04 2005

Keywords

• nonn
• easy

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)

Crossrefs

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

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

Programs

Magma

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

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 *)

SageMath

[lucas_number1(2*n+2,2,-1) -(-1)^n for n in (0..30)] # G. C. Greubel, Aug 21 2022

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)