On the number of odd binomial coefficients
type: Item
instance of: scholarly article
title: On the number of odd binomial coefficients
zbMATH Open document ID: 0869.11015
DOI: 10.1007/BF00052108
author: Gerhard Larcher
published in: Acta Mathematica Hungarica
publication date: 1997-09-01T00:00:00Z
review text: Let \(F(N)\) denote the sum-of-digit function \(\sum_{n<N} 2^{s(n)}\), where \(s(n)\) is the sum of the digits of the base 2-expansion of \(n\). The function \(F(N)\) counts the number of odd trinomial coefficients in the first \(N\) rows of Pascal's triangle. The behaviour of \({F(N)\over N^\theta}\), where \(\theta= {\log 3 \over \log 2}\), is described by a 1-periodic continuous function \(f\) of the dyadic logarithm of \(N\). The paper under review goes deeper into some properties of this classical function. The function \(f\) is proved to be differentiable almost everywhere (the values when \(f\) is differentiable are explicitly determined); \(f\) is shown to be nowhere monotonic. Furthermore, the values \(y\) for which the function \(f\) reaches its minimum are proved to be irrational: the binary representation of such a value \(g\) is a Sturmian sequence of angle \(\Theta\). One can thus deduce the transcendence of such a \(y\) from a result of \textit{S. Ferenczi} and \textit{C. Mauduit} [J. Number Theory (to appear)]. Finally, the author defines an algorithm which is conjectured to converge to the minimum of the function \(f\), provided a diophantine approximation condition involving \(\Theta\) is fulfilled.
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 11K31
zbMATH DE Number: 897257
zbMATH Keywords: sum-of-digit function
zbMATH Keywords: number of odd trinomial coefficients
zbMATH Keywords: Pascal's triangle
zbMATH Keywords: Sturmian sequence
zbMATH Keywords: algorithm
reviewed by: Valérie Berthé
MaRDI profile type: Publication
Wikidata QID: Q114234161
cites work: A summation formula related to the binary digits
cites work: Mellin transforms and asymptotics: Digital sums
cites work: Number of Odd Binomial Coefficients
cites work: Exponential sums of sum-of-digit functions
cites work: Power and Exponential Sums of Digital Sums Related to Binomial Coefficient Parity
description: scientific article; zbMATH DE number 897257