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# Lucas

> Compute the nth [Lucas number][lucas-number].

<section class="intro">

The [Lucas numbers][lucas-number] are the integer sequence

<!-- <equation class="equation" label="eq:lucas_sequence" align="center" raw="2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, \ldots" alt="Lucas sequence"> -->

<div class="equation" align="center" data-raw-text="2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, \ldots" data-equation="eq:lucas_sequence">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/lucas/docs/img/equation_lucas_sequence.svg" alt="Lucas sequence">
    <br>
</div>

<!-- </equation> -->

The sequence is defined by the recurrence relation

<!-- <equation class="equation" label="eq:lucas_recurrence_relation" align="center" raw="L_n = \begin{cases}2 & \textrm{if}\ n = 0\\1 & \textrm{if}\ n = 1\\L_{n-1} + L_{n-2} & \textrm{if}\ n > 1\end{cases}" alt="Lucas sequence recurrence relation"> -->

<div class="equation" align="center" data-raw-text="L_n = \begin{cases}2 &amp; \textrm{if}\ n = 0\\1 &amp; \textrm{if}\ n = 1\\L_{n-1} + L_{n-2} &amp; \textrm{if}\ n &gt; 1\end{cases}" data-equation="eq:lucas_recurrence_relation">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/lucas/docs/img/equation_lucas_recurrence_relation.svg" alt="Lucas sequence recurrence relation">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="usage">

## Usage

```javascript
var lucas = require( '@stdlib/math/base/special/lucas' );
```

#### lucas( n )

Computes the nth [Lucas number][lucas-number].

```javascript
var v = lucas( 0 );
// returns 2

v = lucas( 1 );
// returns 1

v = lucas( 2 );
// returns 3

v = lucas( 3 );
// returns 4

v = lucas( 76 );
// returns 7639424778862807
```

If `n > 76`, the function returns `NaN`, as larger [Lucas numbers][lucas-number] cannot be safely represented in [double-precision floating-point format][ieee754].

```javascript
var v = lucas( 77 );
// returns NaN
```

If not provided a nonnegative integer value, the function returns `NaN`.

```javascript
var v = lucas( 3.14 );
// returns NaN

v = lucas( -1 );
// returns NaN
```

If provided `NaN`, the function returns `NaN`.

```javascript
var v = lucas( NaN );
// returns NaN
```

</section>

<!-- /.usage -->

<section class="notes">

</section>

<!-- /.notes -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
var lucas = require( '@stdlib/math/base/special/lucas' );

var v;
var i;

for ( i = 0; i < 77; i++ ) {
    v = lucas( i );
    console.log( v );
}
```

</section>

<!-- /.examples -->

<section class="links">

[lucas-number]: https://en.wikipedia.org/wiki/Lucas_number

[ieee754]: https://en.wikipedia.org/wiki/IEEE_754-1985

</section>

<!-- /.links -->