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

1,1

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.

References

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

Links

Table of n, a(n) for n=1..96.

Eric Weisstein's World of Mathematics, Lagrange Number.

Formula

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

Example

2.9732137494637011045224016...

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A335605 A308320 A254140 * A013500 A244596 A309928

Adjacent sequences: A200988 A200989 A200990 * A200992 A200993 A200994

Keyword

nonn,cons

Author

Alonso del Arte, Dec 06 2011

Status

approved