## Experimental Mathematics

- Experiment. Math.
- Volume 13, Issue 2 (2004), 215-230.

### On a Problem of Steinhaus Concerning Binary Sequences

Shalom Eliahou and Delphine Hachez

#### Abstract

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 *k*th 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.

#### Article information

**Source**

Experiment. Math. Volume 13, Issue 2 (2004), 215-230.

**Dates**

First available in Project Euclid: 20 July 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1090350936

**Mathematical Reviews number (MathSciNet)**

MR2068895

**Zentralblatt MATH identifier**

1070.11008

**Subjects**

Primary: 05A05: Permutations, words, matrices 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx]

Secondary: 11B75: Other combinatorial number theory

**Keywords**

Steinhaus balanced binary sequence derived sequence derived triangle

#### Citation

Eliahou, Shalom; Hachez, Delphine. On a Problem of Steinhaus Concerning Binary Sequences. Experiment. Math. 13 (2004), no. 2, 215--230.https://projecteuclid.org/euclid.em/1090350936