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\begin{center}
\vskip 1cm{\LARGE\bf 
Precious Metal Sequences and \\
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Sierpi\'{n}ski-Type Graphs 
}
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Andreas M. Hinz\footnote{Corresponding author.}\\
Department of Mathematics\\
Ludwig-Maximilians-Universit\"{a}t M\"{u}nchen\\
Theresienstra\3e 39\\
80333 Munich\\
Germany \\
\href{mailto:hinz@math.lmu.de}{\tt 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 \\
\href{mailto:pkstockmeyer@cox.net}{\tt pkstockmeyer@cox.net} 
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\begin{abstract}
 Sierpi\'{n}ski graphs $S_p^n$ and Sierpi\'{n}ski triangle graphs $\ST_p^n$ form two-parametric families of connected simple graphs which are related, for $p=3$, to the Tower of Hanoi with $n$ discs and for $n\rightarrow \infty$ to the Sierpi\'{n}ski triangle fractal. The vertices of minimal degree play a special role as extreme vertices in $S_p^n$ and primitive vertices in $\ST_p^n$. The key concept of this note is that of an $m$-key vertex whose distance to one of the extreme or primitive vertices, respectively, is $m$ times the distance to another one. The number of such vertices and the distances occurring lead to integer sequences with respect to parameter $n$ like, e.g., the Fibonacci sequence (golden) for $p=3$ and the Pell sequence (silver) for $p=4$. The elements of most of these sequences form self-generating sets. We discuss the cases $m=1,2,3,4$ in detail.
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