Carnevale and Voll conjectured that

when

and

are two
distinct integers. We check the conjecture when either

or

is small. We investigate the asymptotic
behaviour of the sum when the ratio

is fixed
and

goes to infinity. We find an explicit range

on which the conjecture is true. We show that the conjecture is almost
surely true for any fixed
r. For
r close to 1, we give several
explicit intervals on which the conjecture is also true.
Received July 3 2019; revised versions received February 5 2020; February 9 2020; May 27 2020.
Published in Journal of Integer Sequences,
June 10 2020.