SINGLE Round Robin Tournament

[Zugzwangle]

Hi all. 
I have a question involving a Round Robin sporting event. I want to create a formula to evenly distribute home and away matches. Firstly, to less complicate the situation, I need only consider even number participants in the tournament, as I would substitute a 'bye' with an extra number if need be.
You can visualise the Round Robin Schedule below.
However I am going to complicate the issue. This is a Single Round robin - so each team plays each other team ONCE only.
From my own mathematical findings, I have found that:
Participants
1). Matches per round = P / 2
2). Rounds = P - 1
3). Games (or I considered this the 'area' of the triangle, and hence a triangle number) , so the nth triangle number =  (P^2 - P) / 2. (Multiply matches per round by Rounds)
4). If I create a Round Robin using a rotating formula such as this where the numbers rotate around the first number (1) the numbers are not correctly distributed between home and away
R1  R2  R3 R4 
12   16  15  etc ....
63   52  41 ....
54   43  32 ....
=>>>> works out as
If H = Home, A=Away
Participant 1 HHHHH
Participant 2 AAHHH
Participant 3 AAHHH
Participant 4 AAHHH
Participant 5 AAHHH
Participant 6 AAHHH

An example of how the finished tournament would look like:

Participant 1 HAHAH
Participant 2 AHAHA
Participant 3 HAHAH
Participant 4 AHAHA
Participant 5 HAHAH
Participant 6 AHAHA

Given I have already paired the participants, how might I go about formulating an equally distributed Single Round Robin Tournament!!

[Zugzwangle]

I have been given this formula.

n = teams / 2...

The teams should only be even numbers. n is always even!!

(1,2,3,4... 2n .... ) which sums up to:
 if (i v j)  ---> 1 < (|i-j| % 2n) < n - 1 ,   i = home,
 if (i v j)  ---> n + 1 < (|i-j| % 2n) < 2n - 1,   i = home, 
if (i v j)  --->  (|i-j| % 2n) = n  and ( i > j) i = home else i = away

 

This produces an array of 12 teams such as: 

Where Rounds --> Matches --> Teams --> Team numbers

Array
(
    [0] => Array
        (
            [0] => Array
                (
                    [0] => 7
                    [1] => 1
                )

            [1] => Array
                (
                    [0] => 8
                    [1] => 2
                )

            [2] => Array
                (
                    [0] => 9
                    [1] => 3
                )

            [3] => Array
                (
                    [0] => 10
                    [1] => 4
                )

            [4] => Array
                (
                    [0] => 11
                    [1] => 5
                )

            [5] => Array
                (
                    [0] => 12
                    [1] => 6
                )

        )

    [1] => Array
        (
            [0] => Array
                (
                    [0] => 2
                    [1] => 1
                )

            [1] => Array
                (
                    [0] => 3
                    [1] => 7
                )

            [2] => Array
                (
                    [0] => 4
                    [1] => 8
                )

            [3] => Array
                (
                    [0] => 5
                    [1] => 9
                )

            [4] => Array
                (
                    [0] => 6
                    [1] => 10
                )

            [5] => Array
                (
                    [0] => 12
                    [1] => 11
                )

        )

    [2] => Array
        (
            [0] => Array
                (
                    [0] => 1
                    [1] => 3
                )

            [1] => Array
                (
                    [0] => 4
                    [1] => 2
                )

            [2] => Array
                (
                    [0] => 5
                    [1] => 7
                )

            [3] => Array
                (
                    [0] => 6
                    [1] => 8
                )

            [4] => Array
                (
                    [0] => 12
                    [1] => 9
                )

            [5] => Array
                (
                    [0] => 11
                    [1] => 10
                )

        )

    [3] => Array
        (
            [0] => Array
                (
                    [0] => 4
                    [1] => 1
                )

            [1] => Array
                (
                    [0] => 5
                    [1] => 3
                )

            [2] => Array
                (
                    [0] => 6
                    [1] => 2
                )

            [3] => Array
                (
                    [0] => 12
                    [1] => 7
                )

            [4] => Array
                (
                    [0] => 11
                    [1] => 8
                )

            [5] => Array
                (
                    [0] => 10
                    [1] => 9
                )

        )

    [4] => Array
        (
            [0] => Array
                (
                    [0] => 1
                    [1] => 5
                )

            [1] => Array
                (
                    [0] => 6
                    [1] => 4
                )

            [2] => Array
                (
                    [0] => 3
                    [1] => 12
                )

            [3] => Array
                (
                    [0] => 2
                    [1] => 11
                )

            [4] => Array
                (
                    [0] => 10
                    [1] => 7
                )

            [5] => Array
                (
                    [0] => 9
                    [1] => 8
                )

        )

    [5] => Array
        (
            [0] => Array
                (
                    [0] => 6
                    [1] => 1
                )

            [1] => Array
                (
                    [0] => 5
                    [1] => 12
                )

            [2] => Array
                (
                    [0] => 4
                    [1] => 11
                )

            [3] => Array
                (
                    [0] => 3
                    [1] => 10
                )

            [4] => Array
                (
                    [0] => 2
                    [1] => 9
                )

            [5] => Array
                (
                    [0] => 8
                    [1] => 7
                )

        )

    [6] => Array
        (
            [0] => Array
                (
                    [0] => 1
                    [1] => 12
                )

            [1] => Array
                (
                    [0] => 11
                    [1] => 6
                )

            [2] => Array
                (
                    [0] => 10
                    [1] => 5
                )

            [3] => Array
                (
                    [0] => 9
                    [1] => 4
                )

            [4] => Array
                (
                    [0] => 8
                    [1] => 3
                )

            [5] => Array
                (
                    [0] => 7
                    [1] => 2
                )

        )

    [7] => Array
        (
            [0] => Array
                (
                    [0] => 11
                    [1] => 1
                )

            [1] => Array
                (
                    [0] => 10
                    [1] => 12
                )

            [2] => Array
                (
                    [0] => 9
                    [1] => 6
                )

            [3] => Array
                (
                    [0] => 8
                    [1] => 5
                )

            [4] => Array
                (
                    [0] => 7
                    [1] => 4
                )

            [5] => Array
                (
                    [0] => 2
                    [1] => 3
                )

        )

    [8] => Array
        (
            [0] => Array
                (
                    [0] => 1
                    [1] => 10
                )

            [1] => Array
                (
                    [0] => 9
                    [1] => 11
                )

            [2] => Array
                (
                    [0] => 8
                    [1] => 12
                )

            [3] => Array
                (
                    [0] => 7
                    [1] => 6
                )

            [4] => Array
                (
                    [0] => 2
                    [1] => 5
                )

            [5] => Array
                (
                    [0] => 3
                    [1] => 4
                )

        )

    [9] => Array
        (
            [0] => Array
                (
                    [0] => 9
                    [1] => 1
                )

            [1] => Array
                (
                    [0] => 8
                    [1] => 10
                )

            [2] => Array
                (
                    [0] => 7
                    [1] => 11
                )

            [3] => Array
                (
                    [0] => 12
                    [1] => 2
                )

            [4] => Array
                (
                    [0] => 3
                    [1] => 6
                )

            [5] => Array
                (
                    [0] => 4
                    [1] => 5
                )

        )

    [10] => Array
        (
            [0] => Array
                (
                    [0] => 1
                    [1] => 8
                )

            [1] => Array
                (
                    [0] => 7
                    [1] => 9
                )

            [2] => Array
                (
                    [0] => 10
                    [1] => 2
                )

            [3] => Array
                (
                    [0] => 11
                    [1] => 3
                )

            [4] => Array
                (
                    [0] => 12
                    [1] => 4
                )

            [5] => Array
                (
                    [0] => 5
                    [1] => 6
                )

        )

)

If displayed as, Teams --> Rounds --> Home / Away.. it looks like this:

Array
(
    [1] => Array
        (
            [0] => 1
            [1] => 1
            [2] => 0
            [3] => 1
            [4] => 0
            [5] => 1
            [6] => 0
            [7] => 1
            [8] => 0
            [9] => 1
            [10] => 0
        )

    [2] => Array
        (
            [0] => 1
            [1] => 0
            [2] => 1
            [3] => 1
            [4] => 0
            [5] => 0
            [6] => 1
            [7] => 0
            [8] => 0
            [9] => 1
            [10] => 1
        )

    [3] => Array
        (
            [0] => 1
            [1] => 0
            [2] => 1
            [3] => 1
            [4] => 0
            [5] => 0
            [6] => 1
            [7] => 1
            [8] => 0
            [9] => 0
            [10] => 1
        )

    [4] => Array
        (
            [0] => 1
            [1] => 0
            [2] => 0
            [3] => 0
            [4] => 1
            [5] => 0
            [6] => 1
            [7] => 1
            [8] => 1
            [9] => 0
            [10] => 1
        )

    [5] => Array
        (
            [0] => 1
            [1] => 0
            [2] => 0
            [3] => 0
            [4] => 1
            [5] => 0
            [6] => 1
            [7] => 1
            [8] => 1
            [9] => 1
            [10] => 0
        )

    [6] => Array
        (
            [0] => 1
            [1] => 0
            [2] => 0
            [3] => 0
            [4] => 0
            [5] => 0
            [6] => 1
            [7] => 1
            [8] => 1
            [9] => 1
            [10] => 1
        )

    [7] => Array
        (
            [0] => 0
            [1] => 1
            [2] => 1
            [3] => 1
            [4] => 1
            [5] => 1
            [6] => 0
            [7] => 0
            [8] => 0
            [9] => 0
            [10] => 0
        )

    [8] => Array
        (
            [0] => 0
            [1] => 1
            [2] => 1
            [3] => 1
            [4] => 1
            [5] => 0
            [6] => 0
            [7] => 0
            [8] => 0
            [9] => 0
            [10] => 1
        )

    [9] => Array
        (
            [0] => 0
            [1] => 1
            [2] => 1
            [3] => 1
            [4] => 0
            [5] => 1
            [6] => 0
            [7] => 0
            [8] => 0
            [9] => 0
            [10] => 1
        )

    [10] => Array
        (
            [0] => 0
            [1] => 1
            [2] => 1
            [3] => 0
            [4] => 0
            [5] => 1
            [6] => 0
            [7] => 0
            [8] => 1
            [9] => 1
            [10] => 0
        )

    [11] => Array
        (
            [0] => 0
            [1] => 1
            [2] => 0
            [3] => 0
            [4] => 1
            [5] => 1
            [6] => 0
            [7] => 0
            [8] => 1
            [9] => 1
            [10] => 0
        )

    [12] => Array
        (
            [0] => 0
            [1] => 0
            [2] => 0
            [3] => 0
            [4] => 1
            [5] => 1
            [6] => 1
            [7] => 1
            [8] => 1
            [9] => 0
            [10] => 0
        )

)

As you can see, although there ARE a fairly distributed number of teams with home or away, in some cases there are long strings of matches at home, for some teams!!

Can you help me sort this one out!! 

Please try if you can. 
 

Thanks in advance