Catalan Numbers, Lucas numbers, and Circuits

Abstract

Shapiro [8] asked what simple family of circuits will have resistances $C_{2n}/C_{2n}-1$ (or something similar) where $C_m=\frac{1}{m+1}(\genfrac{}{}{0ex}{}{2m}{m})$ is the $m$th Catalan number. In this paper, we give a construction of such circuits; we also discuss some related problems.