A123482 Coefficients of the series giving the best rational approximations to sqrt(11).

60, 23940, 9528120, 3792167880, 1509273288180, 600686976527820, 239071907384784240, 95150018452167599760, 37869468272055319920300, 15071953222259565160679700, 5998599512991034878630600360, 2387427534217209622129818263640, 950190160018936438572789038328420

Offset

1,1

Comments

The partial sums of the series 10/3 - 1/a(1) - 1/a(2) - 1/a(3) - ... give the best rational approximations to sqrt(11), which constitute every second convergent of the continued fraction. The corresponding continued fractions are [3;3,6,3], [3;3,6,3,6,3], [3;3,6,3,6,3,6,3] and so forth.

Links

G. C. Greubel, Table of n, a(n) for n = 1..380

Index entries for linear recurrences with constant coefficients, signature (399,-399,1).

Formula

a(n+3) = 399*a(n+2) - 399*a(n+1) + a(n).

a(n) = -5/33 + (5/66 + 1/44*11^(1/2))*(199 + 60*11^(1/2))^n + (5/66 - 1/44*11^(1/2))*(199 - 60*11^(1/2))^n.

G.f.: -60*x / ((x-1)*(x^2-398*x+1)). - Colin Barker, Jun 23 2014

Mathematica

CoefficientList[Series[-60*x/((x - 1)*(x^2 - 398*x + 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 13 2017 *)

Prog

(PARI) Vec(-60*x/((x-1)*(x^2-398*x+1)) + O(x^100)) \\ Colin Barker, Jun 23 2014

Crossrefs

Cf. A010468, A041014, A041015.

Cf. A123478, A123479, A123480, A029549.

Sequence in context: A275051 A068295 A177643 * A263606 A220371 A092870

Adjacent sequences: A123479 A123480 A123481 * A123483 A123484 A123485

Keyword

nonn,easy

Author

Gene Ward Smith, Oct 02 2006

Extensions

More terms from Colin Barker, Jun 23 2014

Status

approved