Coxeter's Circles/Spheres

[Antreas P. Hatzipolakis]

ID Number: A014217
Sequence:  1,1,2,4,6,11,17,29,46,76,122,199,321,521,842,1364,2206,3571,
           5777,9349,15126,24476,39602,64079,103681,167761,271442,
           439204,710646,1149851,1860497,3010349,4870846,7881196,
           12752042,20633239,33385282
Name:      [ ((1+sqrt(5))/2)^n ].
Keywords:  nonn
Offset:    0
Author(s): Clark Kimberling (ck6 at cedar.evansville.edu)

          http://www.research.att.com/~njas/sequences/index.html
__________________________________________________________________

More integral-part-sequences:

H.S.M. Coxeter: Loxodromic Sequences of Tangent Spheres.
Aequationes Mathematicae 1 (1968) 104 - 121

CIRCLES:
Let r_1, r_2,...., r_n,.... be the radii of a sequence of circles so that:
the radii are in geometric progression, and every four consecutive circles
are mutually tangent. The ratio of the g. progression is

                            phi +- sqrt(phi)

where phi is the golden number: (1 + sqrt(5))/2

We can prove it easily by applying Descartes - Soddy Formula:

2(1/(r_1)^2 + 1/(r_2)^2 + 1/(r_3)^2 + 1/(r_4)^2) =
(1/r_1 + 1/r_2 + 1/r_3 + 1/r_4)^2   [1]

Assume that r_1 = 1, then r_2 = k, r_3 = k^2, r_4 = k^3, and [1] becomes:

(k^2 + 1)*(k^3 - 2k^3 - 2k^2 - 2k + 1) = 0

The real roots of this equation are: phi +- sqrt(phi)
(+ for increasing, - for decreasing seq.)

phi + sqrt(phi) = 2.89005363....

Now, the sequence [phi + sqrt(phi)]^n (n>=0) is:
1,2,8,24,68,201,582,1683,4866....

SPHERES:
Let R_1, R_2,...., R_n,.... be the radii of a sequence of spheres so that:
the radii are in geometric progression, and every five consecutive spheres
are mutually tangent.

<q>
John Robinson's sculpture FIRMAMENT is based on seven such spheres
whose radii are in geometric progression;
</q>
        http://www.bangor.ac.uk/SculMath/image/donald.htm

The ratio is:  1/2 * (sqrt(2) + 1 + sqrt(2*sqrt(2) - 1)) = 1.8832........

And the sequence [1/2 * (sqrt(2) + 1 + sqrt(2*sqrt(2) - 1))] ^ n  (n>=0) is:
1,1,3,6,12,23,44,83,158,.....

Note: See also:
%N A045626 Bends in loxodromic sequence of spheres in which each 5 consecutive
spheres are in mutual contact.
%N A027674 Numerical distance between mth and (m+n)th spheres in loxodromic
sequence of spheres in which each 5 consecutive spheres are in mutual contact.
%N A045821 Numerical distance between mth and (n+m)th circles in a loxodromic
sequence of circles in which each 4 consecutive circles touch.
       http://www.research.att.com/~njas/sequences/index.html


Antreas