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

Author

Gene Ward Smith, Oct 02 2006

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A010468, A041014, A041015.

Cf. A123478, A123479, A123480, A029549.

Programs

Mathematica

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

PARI

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

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

Extensions

More terms from Colin Barker, Jun 23 2014