int year = 2016; // 366 days in 2016 because leap year
DateTime yearStart = new DateTime(year, 1, 1);
int daysInYear = (yearStart.AddYears(1) - yearStart).Days; // 366
var oddSaturdays = Enumerable.Range(0, daysInYear)
    .Select(day => yearStart.AddDays(day))
    .Where(date => date.Day % 2 == 1 && date.DayOfWeek == DayOfWeek.Saturday);

foreach (DateTime oddSaturday in oddSaturdays)
{ 
    // ...
    string sql = string.Format("INSERT INTO {0}.[WEEKEND] ([DATE],[DAYNAME]) VALUES (@DATE, @DAYNAME)", schemaname);
    using (var cmd = new SqlCommand(sql, con))
    {
        cmd.Parameters.Add("@DATE", SqlDbType.Date).Value = oddSaturday;
        cmd.Parameters.Add("@DAYNAME", SqlDbType.VarChar).Value = oddSaturday.DayOfWeek.ToString();
        int inserted = cmd.ExecuteNonQuery();
    }
    // ...
}