Kevin Ryde > Math-NumSeq-69 > Math::NumSeq::Pell

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# CPAN RT

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Module Version: 69

# 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))```

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