

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
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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 / ((x1)*(x^298*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/((x1)*(x^298*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



