This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A123480 #28 Sep 16 2024 23:58:10
%S A123480 4,60,840,11704,163020,2270580,31625104,440480880,6135107220,
%T A123480 85451020204,1190179175640,16577057438760,230888624967004,
%U A123480 3215863692099300,44791203064423200,623860979209825504,8689262505873133860,121025814103014048540,1685672134936323545704
%N A123480 Coefficients of the series giving the best rational approximations to sqrt(3).
%C A123480 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 A123480 G. C. Greubel, <a href="/A123480/b123480.txt">Table of n, a(n) for n = 1..500</a>
%H A123480 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (15,-15,1).
%F A123480 a(n+3) = 15*a(n+2) - 15*a(n+1) + a(n).
%F A123480 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 A123480 a(n) = 4*A076139(n) = 2*A217855(n) = 1/2*A045899(n) = 4/3*A076140(n). - _Peter Bala_, Dec 31 2012
%F A123480 G.f.: -4*x/((x-1)*(x^2-14*x+1)). - _Colin Barker_, Jan 20 2013
%F A123480 a(n) = A001353(n)*A001353(n+1). - _Antonio Alberto Olivares_, Apr 06 2020
%t A123480 CoefficientList[Series[-4*x/((x - 1)*(x^2 - 14*x + 1)), {x, 0, 50}], x] (* _G. C. Greubel_, Oct 13 2017 *)
%o A123480 (PARI) my(x='x+O('x^50)); Vec(-4*x/((x-1)*(x^2-14*x+1))) \\ _G. C. Greubel_, Oct 13 2017
%Y A123480 Cf. A123478, A123479, A029549, A123482. A045899, A076139, A076140, A217855.
%K A123480 nonn,easy
%O A123480 1,1
%A A123480 _Gene Ward Smith_, Sep 28 2006