Precious Metal Sequences and
Sierpinski-Type Graphs
Andreas M. Hinz
1
Department of Mathematics
Ludwig-Maximilians-Universität München
Theresienstraße 39
80333 Munich
Germany
hinz@math.lmu.de
Paul K. Stockmeyer
Department of Computer Science
The College of William & Mary
P.O. Box 8795
Williamsburg, VA 23187-8795
USA
pkstockmeyer@cox.net
Sierpinski graphs
![$S_p^n$](img2.png)
and Sierpinski triangle graphs
![$\widehat{S}_p^n$](img3.png)
form two-parametric families of connected simple graphs which are related, for
![$p=3$](img4.png)
, to the Tower of Hanoi with
![$n$](img5.png)
discs and for
![$n\rightarrow \infty$](img6.png)
to the Sierpinski triangle fractal. The vertices of minimal degree play a special role as extreme vertices in
![$S_p^n$](img2.png)
and primitive vertices in
![$\widehat{S}_p^n$](img3.png)
. The key concept of this note is that of an
![$m$](img7.png)
-key vertex whose distance to one of the extreme or primitive vertices, respectively, is
![$m$](img7.png)
times the distance to another one. The number of such vertices and the distances occurring lead to integer sequences with respect to parameter
![$n$](img5.png)
like, e.g., the Fibonacci sequence (golden) for
![$p=3$](img4.png)
and the Pell sequence (silver) for
![$p=4$](img8.png)
. The elements of most of these sequences form self-generating sets. We discuss the cases
![$m=1,2,3,4$](img9.png)
in detail.