There are four types of electron orbitals, s, p, d, and f. In each energy level, there are one, three, and five orbitals of the s, p, and d types, respectively. Using the same pattern, how many f orbitals are there in each energy level?

The pattern you speak of that tells us the number of orbitals of each type results from solving of Schroedinger's equation. In the course of solving this equation, one generates quantum numbers that determine the characteristics of the electrons in an atom.

The first of these, called the principle quantum nnumber, #n#, determines which energy level (shell) we are considering. #n#=1 is the first shell, #n#=2 is the second shell and so on.

The second quantum number, called the azimuthal quantum number, #l#, determines the type of orbital (#s, p, d or f#) within that shell. For an s-orbital, #l#=0, for p-orbitals, #l=1# and so on. For any shell, #l# can have values from 0 to #n#.

The numbers you quote in your question result from the third of these quantum numbers, called the magnetic quantum number. Each possible value of this quantum number gives us a unique orbital in a particular shell (energy level). It can have #2l+1# different values for a given value of #l#.

So the mathematical patterns that appear in the quantum mechanical model all have very solid theory behind the values seen.