Can we have singleton class with public constructor?

// non-static Ctor: static internal instance
public class OpenDoorSingleton1
{
    private static OpenDoorSingleton1 _instance;

    private string _aString;

    public string AString
    {
        set { if (_instance != null) _instance._aString = value; }

        get
        {
            if (_instance != null)
            {
                return _instance._aString;
            }
            else
            {
                return string.Empty;
            }
        }
    }

    // hidden parameterless ctor
    protected OpenDoorSingleton1()
    {
    }

    public static OpenDoorSingleton1 StaticInstantiate()
    {
        if (_instance == null) _instance = new OpenDoorSingleton1();

        return _instance;
    }

    public OpenDoorSingleton1(string astring)
    {
        _instance = StaticInstantiate();
        _instance.AString = astring;
    }
}