npm package: @stdlib/math-base-special-lucas
Description: Compute the nth Lucas number.
Author: The Stdlib Authors
Latest version: 0.3.1

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# Lucas

[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] <!-- [![dependencies][dependencies-image]][dependencies-url] -->

> 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="installation">

## Installation

```bash
npm install @stdlib/math-base-special-lucas
```

</section>

<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 -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
#include "stdlib/math/base/special/lucas.h"
```

#### stdlib_base_lucas( n )

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

```c
double out = stdlib_base_lucas( 0.0 );
// returns 2.0

out = stdlib_base_lucas( 1.0 );
// returns 1.0
```

The function accepts the following arguments:

-   **n**: `[in] double` input value.

```c
double stdlib_base_lucas( const double n );
```

</section>

<!-- /.usage -->

<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

<!-- C API usage examples. -->

<section class="examples">

### Examples

```c
#include "stdlib/math/base/special/lucas.h"
#include <stdio.h>

int main( void ) {
    double i;
    double v;

    for ( i = 0.0; i < 77.0; i++ ) {
        v = stdlib_base_lucas( i );
        printf( "lucas(%lf) = %lf\n", i, v );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->

<section class="related">

* * *

## See Also

-   <span class="package-name">[`@stdlib/math-base/special/fibonacci`][@stdlib/math/base/special/fibonacci]</span><span class="delimiter">: </span><span class="description">compute the nth Fibonacci number.</span>
-   <span class="package-name">[`@stdlib/math-base/special/negalucas`][@stdlib/math/base/special/negalucas]</span><span class="delimiter">: </span><span class="description">compute the nth negaLucas number.</span>

</section>

<!-- /.related -->

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<section class="main-repo" >

* * *

## Notice

This package is part of [stdlib][stdlib], a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop [stdlib][stdlib], see the main project [repository][stdlib].

#### Community

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---

## License

See [LICENSE][stdlib-license].


## Copyright

Copyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].

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[stdlib-license]: https://raw.githubusercontent.com/stdlib-js/math-base-special-lucas/main/LICENSE

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

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

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[@stdlib/math/base/special/fibonacci]: https://www.npmjs.com/package/@stdlib/math-base-special-fibonacci

[@stdlib/math/base/special/negalucas]: https://www.npmjs.com/package/@stdlib/math-base-special-negalucas

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</section>

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