A200991 Decimal expansion of square root of 221/25.

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, 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, 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

Offset: 1

Author

Alonso del Arte, Dec 06 2011

Keywords

• nonn
• cons

Comments

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.

Continued fraction is 2 followed by 1, 36, 3, 148, 3, 36, 1, 4 repeated.

Examples

			2.9732137494637011045224016...
		

References

• J. H. Conway and R. K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996, p. 187

Links

• Eric Weisstein's World of Mathematics, Lagrange Number.

Crossrefs

Cf. A002163 (the first Lagrange number), A010466 (the second Lagrange number).

Programs

Mathematica

RealDigits[Sqrt[221/25], 10, 100][[1]]

PARI

sqrt(221)/5 \\ Charles R Greathouse IV, Dec 06 2011

Formula

With m = 5 being a Markov number (A002559), L = sqrt(9 - 4/m^2).