Kevin Ryde > Math-NumSeq-71 > Math::NumSeq::Pell

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Module Version: 71   Source  

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[k-1] + P[k-2] 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 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.

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 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.

FORMULAS ^

Value to i Estimate

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))

SEE ALSO ^

Math::NumSeq, Math::NumSeq::Fibonacci, Math::NumSeq::LucasNumbers

HOME PAGE ^

http://user42.tuxfamily.org/math-numseq/index.html

LICENSE ^

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/>.

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