Near Noble Number

A Real Number $0<\nu<1$ whose Continued Fraction is periodic, and the periodic sequence of terms is composed of a string of 1s followed by an Integer $n>1$,

$$\nu =[\,\overline{\underbrace{1, 1, \ldots, 1}_P, n}\,].$$ (1)

This can be written in the form

$$\nu = [\underbrace{1, 1, \ldots, 1}_P, n, \nu^{-1}],$$ (2)

which can be solved to give

$$\nu ={\textstyle{1\over 2}}n\left({\sqrt{1+4{nF_{P-1}+F_{P-2}\over n^2F_P}}-1}\right),$$ (3)

where $F_n$ is a Fibonacci Number. The special case $n=2$ gives

$$\nu=\sqrt{F_{P+2}\over F_P}-1.$$ (4)

See also Noble Number

References

Schroeder, M. R. Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, 2nd enl. ed., corr. printing. Berlin: Springer-Verlag, 1990.

Schroeder, M. ``Noble and Near Noble Numbers.'' In Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman, pp. 392-394, 1991.