++ed by:
Kevin Ryde
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))