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# isSquareTriangularNumber

> Test if a value is a [square triangular number][square-triangular-number].

<section class="intro">

A **square triangular number** is an integer value which is both a [square number][@stdlib/assert/is-square-number] and a [triangular number][@stdlib/assert/is-triangular-number].

[Triangular numbers][@stdlib/assert/is-triangular-number] can be computed using the following formula

<!-- <equation class="equation" label="eq:triangular_number" align="center" raw="T_n = \frac{n(n+1)}{2}" alt="Triangular number formula."> -->

<div class="equation" align="center" data-raw-text="T_n = \frac{n(n+1)}{2}" data-equation="eq:triangular_number">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@b295a09a80f4fd0cc84682dcda0fe3e354394c0c/lib/node_modules/@stdlib/assert/is-square-triangular-number/docs/img/equation_triangular_number.svg" alt="Triangular number formula.">
    <br>
</div>

<!-- </equation> -->

for `n >= 0`.

By analogy with the square root of `x`, one can define the positive triangular root of `x` such that `T_n = x`

<!-- <equation class="equation" label="eq:triangular_root" align="center" raw="n = \frac{\sqrt{8x+1} - 1}{2}" alt="Triangular root formula."> -->

<div class="equation" align="center" data-raw-text="n = \frac{\sqrt{8x+1} - 1}{2}" data-equation="eq:triangular_root">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@b295a09a80f4fd0cc84682dcda0fe3e354394c0c/lib/node_modules/@stdlib/assert/is-square-triangular-number/docs/img/equation_triangular_root.svg" alt="Triangular root formula.">
    <br>
</div>

<!-- </equation> -->

Accordingly, an integer `x` is a [triangular number][@stdlib/assert/is-triangular-number] **if and only** if `8x+1` is a [square number][@stdlib/assert/is-square-number].

</section>

<!-- /.intro -->

<section class="usage">

## Usage

```javascript
var isSquareTriangularNumber = require( '@stdlib/assert/is-square-triangular-number' );
```

#### isSquareTriangularNumber( value )

Tests if a `value` is a [square triangular number][square-triangular-number].

<!-- eslint-disable no-new-wrappers -->

```javascript
var Number = require( '@stdlib/number/ctor' );

var bool = isSquareTriangularNumber( 36.0 );
// returns true

bool = isSquareTriangularNumber( new Number( 36.0 ) );
// returns true

bool = isSquareTriangularNumber( 3.14 );
// returns false

bool = isSquareTriangularNumber( -5.0 );
// returns false

bool = isSquareTriangularNumber( NaN );
// returns false

bool = isSquareTriangularNumber( null );
// returns false
```

#### isSquareTriangularNumber.isPrimitive( value )

Tests if a `value` is a primitive [square triangular number][square-triangular-number].

<!-- eslint-disable no-new-wrappers -->

```javascript
var Number = require( '@stdlib/number/ctor' );

var bool = isSquareTriangularNumber.isPrimitive( 36.0 );
// returns true

bool = isSquareTriangularNumber.isPrimitive( new Number( 36.0 ) );
// returns false
```

#### isSquareTriangularNumber.isObject( value )

Tests if a `value` is a `Number` object having a value which is a [square triangular number][square-triangular-number].

<!-- eslint-disable no-new-wrappers -->

```javascript
var Number = require( '@stdlib/number/ctor' );

var bool = isSquareTriangularNumber.isObject( 36.0 );
// returns false

bool = isSquareTriangularNumber.isObject( new Number( 36.0 ) );
// returns true
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   Return values are not reliable for numbers greater than `1125899906842624`.

</section>

<!-- /.notes -->

<section class="examples">

## Examples

<!-- eslint-disable no-new-wrappers -->

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

```javascript
var Number = require( '@stdlib/number/ctor' );
var isSquareTriangularNumber = require( '@stdlib/assert/is-square-triangular-number' );

var bool = isSquareTriangularNumber( 36.0 );
// returns true

bool = isSquareTriangularNumber( new Number( 36.0 ) );
// returns true

bool = isSquareTriangularNumber( 0.0 );
// returns true

bool = isSquareTriangularNumber( 1.0 );
// returns true

bool = isSquareTriangularNumber( 3.14 );
// returns false

bool = isSquareTriangularNumber( -5.0 );
// returns false

bool = isSquareTriangularNumber( NaN );
// returns false

bool = isSquareTriangularNumber( '0.5' );
// returns false

bool = isSquareTriangularNumber( null );
// returns false
```

</section>

<!-- /.examples -->

<section class="links">

[square-triangular-number]: https://en.wikipedia.org/wiki/Square_triangular_number

[@stdlib/assert/is-square-number]: https://github.com/stdlib-js/assert/tree/main/is-square-number

[@stdlib/assert/is-triangular-number]: https://github.com/stdlib-js/assert/tree/main/is-triangular-number

</section>

<!-- /.links -->
