A123479 Coefficients of series giving the best rational approximations to sqrt(6).

20, 1980, 194040, 19013960, 1863174060, 182572043940, 17890197132080, 1753056746899920, 171781670999060100, 16832850701160989900, 1649447587042777950120, 161629030679491078121880, 15837995559003082877994140, 1551961935751622630965303860

Offset

1,1

Comments

The partial sums of the series 5/2 - 1/a(1) - 1/a(2) - 1/a(3) - ... give the best rational approximations to sqrt(6), which constitute every second convergent of the continued fraction. The corresponding continued fractions are [2;2], [2;2,4,2], [2;2,4,2,4,2], [2;2,4,2,4,2,4,2] and so forth.

Sequence of numbers x=a(n) such 4*x+1 and 6*x+1 are both square, and their square roots are A138288(n) and A054320(n). - Paul Cleary, Jun 23 2014

Links

G. C. Greubel, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Formula

a(n+3) = 99*a(n+2) - 99*a(n+1) + a(n).

a(n) = -5/24 + (( + 2*6^(1/2))/48)*(49 + 20*6^(1/2))^n + ((5 - 2*6^(1/2))/48)*(49 - 20*6^(1/2))^n.

G.f.: -20*x / ((x-1)*(x^2-98*x+1)). - Colin Barker, Jun 23 2014

Mathematica

LinearRecurrence[{99, -99, 1}, {0, 20, 1980}, {2, 25}] (* Paul Cleary, Jun 23 2014 *)

Prog

(PARI) Vec(-20*x/((x-1)*(x^2-98*x+1)) + O(x^100)) \\ Colin Barker, Jun 23 2014

Crossrefs

Cf. A123478, A123480, A029549, A123482.

Sequence in context: A222973 A267575 A072818 * A071152 A195622 A305658

Adjacent sequences: A123476 A123477 A123478 * A123480 A123481 A123482

Keyword

nonn,easy

Author

Gene Ward Smith, Sep 28 2006

Extensions

More terms from Colin Barker, Jun 23 2014

Status

approved