What a picket fence calculator solves
The fetched Omni fence picket page isolates the picket-count question from the rest of the fence. Once you know the fence length, picket width, and picket spacing, the required number of pickets follows directly.
Picket-count formula from Omni
TP = FL / (PW + PS)
In Omni's notation, TP is total pickets, FL is fence length, PW is picket width, and PS is picket spacing. The result should be rounded up to the next whole picket.
Source example
Omni's FAQ gives a 200 ft fence with 5 in pickets and 1 in spacing. Converting 200 ft to 2,400 in and dividing by 6 in gives 400 pickets.