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Binary signed-digit integers and the Stern diatomic sequence

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Abstract

Stern’s diatomic sequence is a well-studied and simply defined sequence with many fascinating characteristics. The binary signed-digit representation of integers is an alternative representation of integers with much use in efficient computation, coding theory and cryptography. We link these two ideas here, showing that the number of i-bit binary signed-digit representations of an integer n with \(n<2^i\) is the \((2^i-n)^{\text {th}}\) element in Stern’s diatomic sequence. This correspondence makes the vast range of results known for Stern’s diatomic sequence available for consideration in the study of binary signed-digit integers.

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Acknowledgements

The author wishes to thank Vanessa Job for useful discussions. The author would also like to thank the anonymous referees for their careful readings and helpful suggestions.

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  1. Ultrascale Systems Research Center, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    Laura Monroe

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  1. Laura Monroe
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Correspondence to Laura Monroe.

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Communicated by J. Jedwab.

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This work was supported by the U.S. Department of Energy through the Los Alamos National Laboratory, and by LANL’s Ultrascale Systems Research Center at the New Mexico Consortium (Contract No. DE-FC02-06ER25750). Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy (Contract No. 89233218CNA000001). The United States Government retains and the publisher, by accepting this work for publication, acknowledges that the United States Government retains an irrevocable, nonexclusive, royalty-free license to publish, translate, reproduce, use, or dispose of the published form of the work and to authorize others to do the same for U.S. Government purposes.

This publication has been assigned the LANL identifier LA-UR-21-25242.

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Monroe, L. Binary signed-digit integers and the Stern diatomic sequence . Des. Codes Cryptogr. 89, 2653–2662 (2021). https://doi.org/10.1007/s10623-021-00903-6

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