Is this the right way to schedule a seeded tournament?

I was curious as to how seeded tournaments are set up. As you probably know, these schedules are set up so that top seeded competitors cannot face each other until the final rounds. The top two seeds can’t meet until the finals, none of the top four seeds can meet until the semi-finals, and so on.

I could not find an algorithm, but I found Web sites that create schedules for you. Does this seem right? Link?

I came up with my own scheduling algorithm, which satisfies the seeding requirement, but which I think is better. My algorithm is simple to describe, but a little clunky to do, since it takes several repetitions. The computer program for this is only a few lines.

First write the seeds in order:
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16

Then move the numbers in even positions to the right to get two sub-tournaments:
(1,3,5,7,9,11,13,15), (2,4,6,8,10,12,14,16)

In the same way, each of the two sub-tournaments are divided into
two sub-tournaments, and so on.

The final result is:
(1, 9), (5, 13), (3, 11), (7, 15), (2, 10), (6, 14), (4, 12), (8, 16)
The schedule at the computer site has the first and last seeds facing each other and the second and next to last seeds facing each other in the first round. That does not seem very fair. My algorithm spreads things out. Each of the initial matches has the competitor seeds separated by 8.