I am trying to compute the number of classes of real vector bundles over spheres. I am reading Bott Tu (Differential Forms in Algebraic Topology) and have worked out using Bott Periodicity Theorem the following table (which I upload as a picture cause I was not able to make the code work here):
Now, using standard stufff from Bott Tu, to be precise: $$\text{Vect}_k(\mathbb{S}^q) \simeq \pi_{q-1}(O(k))/\mathbb{Z}_2.$$
I obtain that the number of classes of real vector bundles over the sphere $\mathbb{S}^q$ are:
I even guess the following, but I am not sure:
Are my results correct? Is there any reference where I can check them?
Thanks in advance.