# Pell numbers

```ruby
func pell_number(n) {
    lucasU(2, -1, n)
}

func pell_lucas_number(n) {
    lucasV(2, -1, n)
}

say ("The first 10 Pell numbers: ", 10.of(pell_number).join(", "))
say ("The first 10 Pell-Lucas numbers: ", 10.of(pell_lucas_number).join(", "))

say "\nFirst 10 rational approximations to √2:"
{|n| pell_lucas_number(n) / 2 / pell_number(n) }.map(1..10).each {|r|
    say "#{'%10s' % r.as_frac} =~ #{r.as_float}"
}

var pell_prime_indices = 10.by {|n| pell_number(n).is_prime }
say "\nThe first 10 Pell primes: "
pell_prime_indices.each {|n|
    say "Pell(#{'%2s' % n}) = #{pell_number(n)}"
}

say ("\nThe first 10 Newman-Shank-Williams numbers: ",
    10.of{|n| pell_number(2*n) + pell_number(2*n + 1) }.join(', '))

say "\nThe first 10 Pythagorean triples corresponding to near isosceles right triangles:"
10.of{|n| pell_number(2*n + 1) }.map {|h|
    var t = isqrt(h**2 >> 1)
    assert_eq(t**2 + (t+1)**2, h**2)
    [t, t+1, h]
}.each{.say}
```

#### Output:

```
The first 10 Pell numbers: 0, 1, 2, 5, 12, 29, 70, 169, 408, 985
The first 10 Pell-Lucas numbers: 2, 2, 6, 14, 34, 82, 198, 478, 1154, 2786

First 10 rational approximations to √2:
       1/1 =~ 1
       3/2 =~ 1.5
       7/5 =~ 1.4
     17/12 =~ 1.41666666666666666666666666666666666666666666667
     41/29 =~ 1.41379310344827586206896551724137931034482758621
     99/70 =~ 1.41428571428571428571428571428571428571428571429
   239/169 =~ 1.41420118343195266272189349112426035502958579882
   577/408 =~ 1.4142156862745098039215686274509803921568627451
  1393/985 =~ 1.41421319796954314720812182741116751269035532995
 3363/2378 =~ 1.41421362489486963835155592935239697224558452481

The first 10 Pell primes: 
Pell( 2) = 2
Pell( 3) = 5
Pell( 5) = 29
Pell(11) = 5741
Pell(13) = 33461
Pell(29) = 44560482149
Pell(41) = 1746860020068409
Pell(53) = 68480406462161287469
Pell(59) = 13558774610046711780701
Pell(89) = 4125636888562548868221559797461449

The first 10 Newman-Shank-Williams numbers: 1, 7, 41, 239, 1393, 8119, 47321, 275807, 1607521, 9369319

The first 10 Pythagorean triples corresponding to near isosceles right triangles:
[0, 1, 1]
[3, 4, 5]
[20, 21, 29]
[119, 120, 169]
[696, 697, 985]
[4059, 4060, 5741]
[23660, 23661, 33461]
[137903, 137904, 195025]
[803760, 803761, 1136689]
[4684659, 4684660, 6625109]
```


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://trizen.gitbook.io/sidef-lang/programming_tasks/p/pell_numbers.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.