<abstract xmlns="http://www.w3.org/1999/xhtml">Given an integral, increasing, linear-recurrent sequence <italic>A</italic> with initial term 1, the greedy algorithm may be used on the terms of <italic>A</italic> to represent all positive integers. For large classes of recurrences, the average digit sum is known to equal <italic>cA</italic> log <italic>n</italic>+<italic>O</italic>(1), where <italic>cA</italic> is a positive constant that depends on <italic>A</italic>. This asymptotic result is re-proved with an elementary approach for a class of special recurrences larger than, or distinct from, that of former papers. The focus is on the constants <italic>cA</italic> for which, among other items, explicit formulas are provided and minimal values are found, or conjectured, for all special recurrences up to a certain order.</abstract>

Given an integral, increasing, linear-recurrent sequence A with initial term 1, the greedy algorithm may be used on the terms of A to represent all positive integers. For large classes of recurrences, the average digit sum is known to equal cA log n+O(1), where cA is a positive constant that depends on A. This asymptotic result is re-proved with an elementary approach for a class of special recurrences larger than, or distinct from, that of former papers. The focus is on the constants cA for which, among other items, explicit formulas are provided and minimal values are found, or conjectured, for all special recurrences up to a certain order.

On the Constant in the Average Digit Sum for a Recurrence-Based Numeration

Université de Caen, Département de Mathématiques et Mécanique, Campus 2, Blvd Maréchal Juin, 14032 Caen,

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"On the Constant in the Average Digit Sum for a Recurrence-Based Numeration"}

Given an integral, increasing, linear-recurrent sequence A with initial term 1, the greedy algorithm may be used on the terms of A to represent all positive...

On the Constant in the Average Digit Sum for a Recurrence-Based Numeration

Published Online: Jan 13, 2017

Page range: 125 - 150

Received: Aug 12, 2015

Accepted: Nov 24, 2015

DOI: https://doi.org/10.1515/udt-2016-0016

Keywords
numeration, digit sum, average, recurrence

© 2016 Christian Ballot, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

On the Constant in the Average Digit Sum for a Recurrence-Based Numeration

Published Online: Jan 13, 2017

Page range: 125 - 150

Received: Aug 12, 2015

Accepted: Nov 24, 2015

DOI: https://doi.org/10.1515/udt-2016-0016

Keywordsnumeration, digit sum, average, recurrence

© 2016 Christian Ballot, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
numeration, digit sum, average, recurrence