In page 20 of Bailey and Crandall’s *On the Random Character of Constant Expansions*, they give the wonderfully unusual sum,

I didn’t think this was an isolated result so set about to find a generalization. I found its counterpart,

Note that,

We can demystify the sum a bit by splitting the log function into parts. After some algebraic manipulation, we find that the first one becomes,

Thus it can be expressed in the form,

where {} are roots of the same equation, {} are roots of another, and *r* is a rational. The fact that,

was my clue that trigonometric functions may be involved. Define,

then for *p* = 5,

*p* = 7

*p* = 9

with the constants {} easily ascertained as {}, and so on. On the other hand, their counterparts are easier as the exponent has the same subscript as the base. Still defining,

then for *p* = 5,

*p* = 7

*p* = 9

etc.