MHB Creating a Balanced Single Round Robin Tournament Schedule

[zugzwangle]

Hi all.
I have a question involving a Round Robin sporting event, so grab a pen if you're up to the challenge. 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:
If P = 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!

 

[MarkFL]

I moved this topic here as it seems a better fit than Geometry.