It's not a good idea to change state of the class the way you do it here only to return the field as a result:

private int _currentIndex;
private string _outputGrid;

public string CalculateNextGrid(string inputGrid, int gridSize)
{
    _outputGrid = "";
    _currentIndex = 0;

    ...

    return _outputGrid;
}

This is not thread safe. Should you ever want to use a single instance of it with multiple threads they will overwirte each other's work. Most of the fields could be local to the functions making them pure ones and thus much easier to use and to maintain.

[..] a pure function has two useful characteristics:

 

• It has no side effects. The function does not change any variables or the data of any type outside of the function.

 
• It is consistent. Given the same set of input data, it will always return the same output value.

It's not a good idea to change state of the class the way you do it here only to return the field as a result:

private int _currentIndex;
private string _outputGrid;

public string CalculateNextGrid(string inputGrid, int gridSize)
{
    _outputGrid = "";
    _currentIndex = 0;

    ...

    return _outputGrid;
}

This is not thread safe. Should you ever want to use a single instance of it with multiple threads they will overwirte each other's work. Most of the fields could be local to the functions making them pure ones and thus much easier to use and to maintain.

[..] a pure function has two useful characteristics:

• It has no side effects. The function does not change any variables or the data of any type outside of the function.

• It is consistent. Given the same set of input data, it will always return the same output value.