Math::NumSeq::Pell -- Pell numbers
use Math::NumSeq::Pell; my $seq = Math::NumSeq::Pell->new; my ($i, $value) = $seq->next;
The Pell numbers
0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, ... starting i=0
where
P[k] = 2*P[k-1] + P[k-2] starting P[0]=0 and P[1]=1
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 a Math::BigInt
to preserve precision.
$seq->seek_to_i($i)
Move the current sequence position to $i
. The next call to next()
will return $i
and corresponding value.
$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 a Math::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.
The Pell numbers are a Lucas sequence and hence a power
(1+sqrt(2))^i - (1-sqrt(2))^i P[i] = ----------------------------- # exactly 2*sqrt(2)
Since abs(1-sqrt(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))
Math::NumSeq, Math::NumSeq::Fibonacci, Math::NumSeq::LucasNumbers
http://user42.tuxfamily.org/math-numseq/index.html
Copyright 2010, 2011, 2012, 2013, 2014 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 <http://www.gnu.org/licenses/>.