Inductance of Shielded Helical Resonators
This formula may be considered as having two parts: Wheeler's formula for the inductance of a solenoid, and a correction term to account for the shield.

Wheeler's formula [1] states:

$$
L \approx \frac{a^2n^2}{9a+10b} \text{ }\text{ }\mu\text{H}
$$

Where \$a\$ is the coil radius in inches and \$b\$ is the coil length in inches.  It is valid for long coils, in particular when \$b&gt;0.8a\$.  Wheeler's formula may be interpreted as increasing the effective solenoid length, as compared to its physical length, by 90% of the radius [2].

The second part of your formula is a term to account for the effects of the shield.  It appears to be an empirical correction suggested by Hayman [3], who states, "Whilst these factors make no pretense to theoretical form or to the correct division of the effect as between the side and ends of the screen, they do enable a coil to be designed to have a given inductance when screened."

[1] H. A. Wheeler, "Simple Inductance Formulas for Radio Coils," in Proceedings of the Institute of Radio Engineers, vol. 16, no. 10, pp. 1398-1400, Oct. 1928.

[2] T. H. Lee.  "Planar Microwave Engineering: A Practical Guide to Theory, Measurement, and Circuits, Volume 1," pp. 141-142.  Cambridge University Press, Aug 30, 2004.

[3] W. G. Hayman.  "Inductance of Solenoids in Cylindrical Screen Boxes," in Wireless Engineer, vol. 11, no. 127, pp. 190, Apr. 1934.  http://www.americanradiohistory.com/Archive-Experimental%20Wireless/30s/Wireless-Engineer-1934-04.pdf