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A123479 Coefficients of series giving the best rational approximations to sqrt(6). 6
20, 1980, 194040, 19013960, 1863174060, 182572043940, 17890197132080, 1753056746899920, 171781670999060100, 16832850701160989900, 1649447587042777950120, 161629030679491078121880, 15837995559003082877994140, 1551961935751622630965303860 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 2 23:51 EDT 2022. Contains 357230 sequences. (Running on oeis4.)