and 1 contributors

# 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, ...
starting i=0``````

where

``    P(k) = 2*P(k-1) + P(k-2)``

starting from i=0 values 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.

## Random Access

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

Return the `\$i`'th Pell number.

`\$bool = \$seq->pred(\$value)`

Return true if `\$value` is a Pell number.

`\$i = \$seq->value_to_i_estimate(\$value)`

Return an estimate of the i corresponding to `\$value`.

# FORMULAS

## Value to i Estimate

The Pell numbers are a Lucas sequence and are 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 roughly

``````         log(value) + log(2*sqrt(2))
i ~= ---------------------------
log(1+sqrt(2))``````

# HOME PAGE

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

# LICENSE

Copyright 2010, 2011, 2012, 2013 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/>.