C# calculate and visualize the summary of 6+2 rests for the whole year.

Initially, you can start by ignoring needing to know what day of the week you are looking at. Just use a loop and some simple DateTime arithmetic. If I were solving this using DateTime (or DateOnly in a grown up .NET version), I would consider the following:

1. The date we start our calculation from is supplied to us as a parameter
   
2. We are ending when the calculated start date of a shift is greater than 1 year from now
   

Knowing this, I would create a list of DateTime (DateOnly) that will contain the first day off.
In a loop, I would add 6 days to the start date. This date would be added to the list.
The new start date would be 2 days from this date.
Repeat the loop until we have past the end date

If I want to know the start dates of my days off, I simply loop over the list and look at the DayOfWeek. The last day of this time off is the item in the list + 1 day.

This isn't complicated code to write. Try it, and let us know how you get on.

 private void btnRiposi_Click(object sender, System.EventArgs e)
        {
            DateTime startDate = new DateTime(2023, 2, 2);
            DateTime endDate = new DateTime(2023, 12, 31);

            for (double loopDate = startDate.ToOADate(); loopDate <= endDate.ToOADate(); loopDate++)
            {
                DateTime selectedDate;
                selectedDate = DateTime.FromOADate(loopDate);
                rtbRiposi.Text = selectedDate.ToString();
            }
        }
    }
}

@Pete O'Hanlon, I tried this but it only saves me the last rest date in RichTextBox, how do I make a list of all rest in current year?

Why are you doing it so complicated ?
It seams to me like a kind of a Shift-Modell.
So ... you take your StartDate (this is your Shift-begin) and add 6 Days to it (that is your Off-begin). Now you add 2 Days to it and you have your new Shift-begin.
Repeat that until you have reached your End-Date ...
I will not provide any code for that - I think this desciption is enough to start ...
Are you familiar with the Function AddDays for DateTime ?