H. S. M. Coxeter - The Descartes circle theorem and Fibonacci numbers

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H. S. M. Coxeter - The Descartes circle theorem and Fibonacci numbers

H. S. M. COXETER, Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada

The Descartes circle theorem and Fibonacci numbers

The numerical distance between two circles in the Euclidean plane is defined to be the number

$\begin{displaymath}\frac{c^2 - a^2 - b^2}{2ab}, \end{displaymath}$

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 mth 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, D₀ = -1. Since it is 1 when a+b = c so that the circles are externally tangent, D₁ = D₂ = D₃ = 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

$\begin{displaymath}D_n = \sum^{[n/2]}_{\nu=0} {n \choose 2 \nu} f_{n- \nu - 2}, \end{displaymath}$

in terms of binomial coefficients and Fibonacci numbers.

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