In Electrician's link, it is very very interesting that in examples 52 and beyond a little they obtained inductance values from the formulas in units of length: centimeters <har har>. For example, L=654pi centimeters instead of L=654pi nanohenries
Funny too as this error seems to carry over several other examples. Glad we had such people working on papers like this way back then
I haven't had a chance to look through the document in detail, but it looks like a great reference to have handy.
About this, we should caution the OP to be careful about units. There are so many different electromagnetic systems of units. Typically, we use SI in engineering now, but many references use Gaussian, or other units. I assume here they are using the modified Gaussian system where inductance has units of length.
Well the formula you presented was for a cylindrical coil not a flat coil, so naturally i assumed you wanted a coil like that.
When I saw that formula, I felt it was very similar to the typical formula for a single loop, not a solenoid. Then it appears the N^2 was added as an attempt to consider multiple turns. However, I agree with you and The Electrician that there is an issue that the loops can not occupy the same space, so I don't think the formula can be very good for multiple turns. If the dimension of the coil are such that the wire diameter is considered, then loops that are offset from the "ideal" loop will have a big effect on the answer, I would think.