NAME
Math::NumSeq::Pell  Pell numbers
SYNOPSIS
use Math::NumSeq::Pell;
my $seq = Math::NumSeq::Pell>new;
my ($i, $value) = $seq>next;
DESCRIPTION
The Pell numbers
0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, ...
starting i=0
where
P[k] = 2*P[k1] + P[k2] starting P[0]=0 and P[1]=1
FUNCTIONS
See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.
$seq = Math::NumSeq::Pell>new ()

Create and return a new sequence object.
($i, $value) = $seq>next()

Return the next index and value in the sequence.
When
$value
exceeds the range of a Perl unsigned integer the return is aMath::BigInt
to preserve precision. $seq>seek_to_i($i)

Move the current sequence position to
$i
. The next call tonext()
will return$i
and corresponding value.
Random Access
$value = $seq>ith($i)

Return the
$i
'th Pell number.For negative <$i> the sequence is extended backwards as P[i]=P[i+2]2*P[i+1]. The effect is the same numbers but negative at negative even i.
i P[i]   0 0 1 1 2 2 <+ negative at even i 3 5  4 12 <+
When
$value
exceeds the range of a Perl unsigned integer the return is aMath::BigInt
to preserve precision. $bool = $seq>pred($value)

Return true if
$value
occurs in the sequence, so is a positive Pell number. $i = $seq>value_to_i_estimate($value)

Return an estimate of the i corresponding to
$value
. See "Value to i Estimate" below.
FORMULAS
Value to i Estimate
The Pell numbers are a Lucas sequence and hence a power
(1+sqrt(2))^i  (1sqrt(2))^i
P[i] =  # exactly
2*sqrt(2)
Since abs(1sqrt(2)) < 1 that term approaches zero, so taking logs the rest gives i approximately
log(value) + log(2*sqrt(2))
i ~= 
log(1+sqrt(2))
SEE ALSO
Math::NumSeq, Math::NumSeq::Fibonacci, Math::NumSeq::LucasNumbers
HOME PAGE
http://user42.tuxfamily.org/mathnumseq/index.html
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014 Kevin Ryde
MathNumSeq 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.
MathNumSeq 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 MathNumSeq. If not, see <http://www.gnu.org/licenses/>.