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%I #22 Apr 27 2020 10:22:26
%S 4,60,840,11704,163020,2270580,31625104,440480880,6135107220,
%T 85451020204,1190179175640,16577057438760,230888624967004,
%U 3215863692099300,44791203064423200,623860979209825504,8689262505873133860,121025814103014048540,1685672134936323545704
%N Coefficients of the series giving the best rational approximations to sqrt(3).
%C The partial sums of the series 2 - 1/a(1) - 1/a(2) - 1/a(3) - ... give the best rational approximations to sqrt(3), which constitute every second convergent of the continued fraction. The corresponding continued fractions are [1;1,2,1], [1;1,2,1,2,1], [1;1,2,1,2,1,2,1], [1;1,2,1,2,1,2,1,2,1] and so forth.
%H G. C. Greubel, <a href="/A123480/b123480.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 (15,-15,1).
%F a(n+3) = 15*a(n+2) - 15*a(n+1) + a(n).
%F a(n) = -1/3 + (1/6 + 1/12*3^(1/2))*(7 + 4*3^(1/2))^n + (1/6 - 1/12*3^(1/2))*(7 - 4*3^(1/2))^n.
%F a(n) = 4*A076139(n) = 2*A217855(n) = 1/2*A045899(n) = 4/3*A076140(n). - _Peter Bala_, Dec 31 2012
%F G.f.: -4*x/((x-1)*(x^2-14*x+1)). - _Colin Barker_, Jan 20 2013
%F a(n) = A001353(n)*A001353(n+1). - _Antonio Alberto Olivares_, Apr 06 2020
%t CoefficientList[Series[-4*x/((x - 1)*(x^2 - 14*x + 1)), {x, 0, 50}], x] (* _G. C. Greubel_, Oct 13 2017 *)
%o (PARI) x='x+O('x^50); Vec(-4*x/((x-1)*(x^2-14*x+1))) \\ _G. C. Greubel_, Oct 13 2017 *)
%Y Cf. A123478, A123479, A029549, A123482. A045899, A076139, A076140, A217855.
%K nonn,easy
%O 1,1
%A _Gene Ward Smith_, Sep 28 2006
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