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?
Each energy level, beginning with
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,
The second quantum number, called the azimuthal quantum number,
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
So the mathematical patterns that appear in the quantum mechanical model all have very solid theory behind the values seen.