# 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))``````