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%I #41 Jul 29 2021 07:00:59
%S 2,9,7,3,2,1,3,7,4,9,4,6,3,7,0,1,1,0,4,5,2,2,4,0,1,6,4,2,7,8,6,2,7,9,
%T 3,3,0,2,8,9,7,9,7,1,0,2,7,4,4,1,7,2,3,1,2,1,1,2,6,1,8,9,6,2,0,5,0,3,
%U 6,7,4,6,2,9,5,6,2,3,3,5,3,1,7,2,3,1,6,7,2,9,2,0,5,4,7,9
%N Decimal expansion of square root of 221/25
%C This is the third Lagrange number, corresponding to the third Markov number (5). With multiples of the golden ration and sqrt(2) excluded from consideration, the Hurwitz irrational number theorem uses this Lagrange number to obtain very good rational approximations for irrational numbers.
%C Continued fraction is 2 followed by 1, 36, 3, 148, 3, 36, 1, 4 repeated.
%D J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996, p. 187
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LagrangeNumber.html">Lagrange Number</a>.
%F With m = 5 being a Markov number (A002559), L = sqrt(9 - 4/m^2).
%e 2.9732137494637011045224016...
%t RealDigits[Sqrt[221/25], 10, 100][[1]]
%o (PARI) sqrt(221)/5 \\ _Charles R Greathouse IV_, Dec 06 2011
%Y Cf. A002163 (the first Lagrange number), A010466 (the second Lagrange number).
%K nonn,cons
%O 1,1
%A _Alonso del Arte_, Dec 06 2011
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