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%I #31 Oct 14 2017 01:49:46
%S 20,1980,194040,19013960,1863174060,182572043940,17890197132080,
%T 1753056746899920,171781670999060100,16832850701160989900,
%U 1649447587042777950120,161629030679491078121880,15837995559003082877994140,1551961935751622630965303860
%N Coefficients of series giving the best rational approximations to sqrt(6).
%C 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.
%C 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
%H G. C. Greubel, <a href="/A123479/b123479.txt">Table of n, a(n) for n = 1..500</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (99,-99,1).
%F a(n+3) = 99*a(n+2) - 99*a(n+1) + a(n).
%F 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.
%F G.f.: -20*x / ((x-1)*(x^2-98*x+1)). - _Colin Barker_, Jun 23 2014
%t LinearRecurrence[{99,-99,1},{0,20,1980},{2, 25}] (* _Paul Cleary_, Jun 23 2014 *)
%o (PARI) Vec(-20*x/((x-1)*(x^2-98*x+1)) + O(x^100)) \\ _Colin Barker_, Jun 23 2014
%Y Cf. A123478, A123480, A029549, A123482.
%K nonn,easy
%O 1,1
%A _Gene Ward Smith_, Sep 28 2006
%E More terms from _Colin Barker_, Jun 23 2014
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