Repeating Continued Fractions

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The continued fraction expansion of a quadratic irrational is eventually periodic; the converse is also true.


The repeating sequence, the initial part of the continued fraction form, the value, and the first few convergents to that value are shown.


Contributed by: Ed Pegg Jr (December 2008)
Additional contributions by: Andreas Lauschke
Open content licensed under CC BY-NC-SA



The continued fraction expansions of the square roots of the integers that are not perfect squares possess the so-called "palindromic property". That means that their periodic element sequences without the last element are palindromic (i. e., can be reversed), and the last element is twice the initial term. For example, the continued fraction expansion of is {5, {1, 1, 3, 5, 3, 1, 1, 10}}. The sequence 1, 1, 3, 5, 3, 1, 1 is palindromic, and the 10 is two times the initial 5.

The following shows this for the first 200 integers:

DeleteCases[ContinuedFraction@Sqrt@Range@200, {_}]

For more, see A Tail of Two Palindromes.

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