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Using $\\Delta$--Y transformations and identities for the Fibonacci and Lucas numbers we obtain explicit formulae for the resistance distance $r_{G_n}(i,j)$ between any two vertices $i$ and $j$ of $G_n$. To our knowledge $\\{G_n\\}_{n=3}^\\infty$ is the first nontrivial family with diameter going to $\\infty$ for which all resistance distances have been explicitly calculated. 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