Summary: The numerical distance between two circles in the Euclidean plane is defined to be the number \[ {c^2-a^2 -b^2\over 2ab}, \] where \(a\) and \(b\) are their radii while \(c\) is the ordinary distance between their centres. An infinite sequence of circles is defined to be loxodromic if every four consecutive members are mutually tangent. \(D_n\) denotes the numerical distance between the \(m\)th and \((m+n)\)th circles (the same for all \(m)\). Obviously \(D_{-n} =D_n\). Since the numerical distance is \(-1\) when \(a=b\) and \(c=0\) so that the two circles coincide, \(D_0=-1\). Since it is 1 when \(a+b=c\) so that the circles are externally tangent, \(D_1=D_2 =D_3=1\). Any number of further values of \(D_n\) can be determined successively by the recurrence equation \[ D_m+D_{m+4}= 2(D_{m+1}+ D_{m+2}+D_{m+3}). \] There is also an explicit formula (4.2) for \(D_n\) (as a function of the sequential distance \(n)\) in terms of binomial coefficients and Fibonacci numbers.