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2004 On a Problem of Steinhaus Concerning Binary Sequences
Shalom Eliahou, Delphine Hachez
Experiment. Math. 13(2): 215-230 (2004).


A finite {\small $\pm 1$} sequence X yields a binary triangle {\small $\Delta X$} whose first row is X, and whose {\small $(k+1)$}th row is the sequence of pairwise products of consecutive entries of its kth row, for all {\small $k \geq 1$}. We say that X is balanced if its derived triangle {\small $\Delta X$} contains as many +1s as {\small $-$}1s. In 1963, Steinhaus asked whether there exist balanced binary sequences of every length {\small $n \equiv$} 0 or 3 mod 4. While this problem has been solved in the affirmative by Harborth in 1972, we present here a different solution. We do so by constructing strongly balanced binary sequences, i.e., binary sequences of length n all of whose initial segments of length {\small $n-4t$} are balanced, for {\small $0 \leq t \leq n/4$}. Our strongly balanced sequences do occur in every length {\small $n \equiv$} 0 or 3 mod 4. Moreover, we provide a complete classification of sufficiently long strongly balanced binary sequences.


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Shalom Eliahou. Delphine Hachez. "On a Problem of Steinhaus Concerning Binary Sequences." Experiment. Math. 13 (2) 215 - 230, 2004.


Published: 2004
First available in Project Euclid: 20 July 2004

zbMATH: 1070.11008
MathSciNet: MR2068895

Primary: 05A05, 05A15
Secondary: 11B75

Rights: Copyright © 2004 A K Peters, Ltd.


Vol.13 • No. 2 • 2004
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