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A curiosity About (<i>−</i>1)<sup>[</sup> <i> <sup>e</sup> </i> <sup>]</sup> +(<i>−</i>1)<sup>[2</sup> <i> <sup>e</sup> </i> <sup>]</sup> + <i>···</i> +(<i>−</i>1)<sup>[</sup> <i> <sup>Ne</sup> </i> <sup>]</sup>
Abstract Let α be an irrational real number; the behaviour of the sum S N ( α ):= ( − 1) [ α ] +( − 1) [2 α ] + ··· +( − 1) [ Nα ] depends on the continued fraction expansion of α/ 2. Since the continued …