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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

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
in terms of binomial coefficients and Fibonacci numbers.

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