How to organise a robin robin where 8 teams play 8 games with each team playing each other team once plus one team twice with only 4 games at a time. I am organising a sports day with 8 teams (T1-T8) and 8 activities (A1-A8) over 8 time slots (t1-t8). Each team can only do each activity once and can only play every other team once expect for repeating 1 fixture (out of necessity). Is this possible? And how would this be formatted in an grid looking something like the image linked below? Obviously each team can only be doing 1 activity at a time.
Sample Grid
 A: Here is a schedule that works:




${}$
T0
T1
T2
T3
T4
T5
T6
T7




A0
0/2
5/6

1/4
3/7





A1

1/3
6/7

2/5
4/0




A2


2/4
7/0

3/6
5/1



A3



3/5
0/1

4/7
6/2


A4
7/3



4/6
1/2

5/0


A5
6/1
0/4



5/7
2/3



A6

7/2
1/5



6/0
3/4


A7
4/5

0/3
2/6



7/1




This schedule can be described concisely as follows:

Number the activities, rounds, and people from $0$ to $7$. 
For each $i\in \{0,\dots,7\}$, the $i^\text{th}$ activity is played by

*

*teams number $i+\color{red}0$ and $i+\color{red}2$ in round $i+\color{blue}0$,

*teams number $i+\color{red}5$ and $i+\color{red}6$ in round $i+\color{blue}1$,

*teams number $i+\color{red}1$ and $i+\color{red}4$ in round $i+\color{blue}3$,

*teams number $i+\color{red}3$ and $i+\color{red}7$ in round $i+\color{blue}4$,

where all addition is modulo $8$.

I got the idea to look for a schedule with this special form from Anders Kaseorg's solution to this related problem, and I just wrote a computer program to check all possibilities for the red and blue parameters until it found one that produced a valid schedule.
