PROBLEM: Closed Form Expansions of Continued Fractions

DESCRIPTION: Let x equal the limit (as n goes to infinity) of g_{n+1}/g_{n},
where g_{n+1} = A g_{n} + B g_{n-1}. Guided by numerical calculations, for
many A and B closed form expressions are derived for the quadratic irrationals
x and x^k for many choices of A and B.

INVESTIGATOR: Dan Fishman

PAPERS:  Dan4.tex   Dan4.dvi   Dan4.pdf

PROGRAMS: dan.nb   2g(i-1)+g(i-2).nb   3g(i-1)+g(i-2).nb   4g(i-1)+g(i-2).nb   7g(i-1)+g(i-2).nb   juniorpaper1.nb

In Spring 2003, a more detailed investigation was performed:

PAPERS:  dfJP2.tex   dfJP2.dvi   dfJP2.pdf

PROGRAMS: Higher Powers.nb   ldivm.nb

Continued Fractions of Repeating Block Length 3, and Their Powers (m=1, l=i^2).nb
Continued Fractions of Repeating Block Length 3, and Their ODD Powers (m=2l-1).nb
Continued Fractions of Repeating Block Length 3, and Their EVEN Powers (m=2l-1).nb
PeriodicCF_and_Kuzmin.nb

IMAGES: Go to http://www.math.princeton.edu/~mathlab/jr02fall/Closed/spring/ and you'll see all images