Math::NumSeq::SqrtContinued -- continued fraction expansion of a square root


 use Math::NumSeq::SqrtContinued;
 my $seq = Math::NumSeq::SqrtContinued->new (sqrt => 2);
 my ($i, $value) = $seq->next;


This is terms in the continued fraction expansion of a square root. It approaches the root by

   sqrt(S) = a[0] + ----------- 
                    a[1] +   1
                           a[2] +   1
                                  a[3] + ...

The first term a[0] is the integer part of the root, leaving a remainder 0 < r < 1 which is expressed as r=1/R with R > 1

   sqrt(S) = a[0] + ---

Then a[1] is the integer part of that R, and so on recursively.

Values a[1] onwards are always a fixed-period repeating sequence. For example sqrt(14) is a[0]=3 and then 1,2,1,6 repeating. For some roots a single value repeats. For example sqrt(2) is a[0]=1 then 2 repeating. See SqrtContinuedPeriod for just the length of the period.


See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.

$seq = Math::NumSeq::SqrtContinued->new (sqrt => $s)

Create and return a new sequence object giving the Continued expansion terms of sqrt($s).

$value = $seq->ith ($i)

Return the i'th term in the continued fraction, starting from i=0 for the integer part of the sqrt.

$i = $seq->i_start ()

Return 0, the first term in the sequence being i=0.


Math::NumSeq, Math::NumSeq::SqrtContinuedPeriod, Math::NumSeq::SqrtDigits, Math::NumSeq::SqrtEngel




Copyright 2011, 2012, 2013, 2014, 2016, 2019, 2020 Kevin Ryde

Math-NumSeq is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.

Math-NumSeq is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Math-NumSeq. If not, see <>.