Fun of Thinking — Math Problems from Around the World

A sequence $a_1a_2\cdots a_n\cdots$ with $a_i \in \{0, 1\}$ is a $0$ - $1$ periodic sequence if there is a positive integer $m$ with $a_{i+m} = a_i$ for all $i$ ; the least such $m$ is its period. For a sequence of period $m$ , an important quality measure is

C(k) = \frac{1}{m}\sum_{i=1}^{m} a_i a_{i+k}, \qquad k = 1, 2, \dots, m-1.

For each of the following period- $5$ sequences, determine whether

C(k) \leqslant \frac{1}{5} \quad \text{for } k = 1, 2, 3, 4:

1. $11010\cdots$ ; 2. $11011\cdots$ ; 3. $10001\cdots$ ; 4. $11001\cdots$ .

Difficulty	6/10
Topics	combinatorics, sequences, Casework

Autocorrelation of $0$ - $1$ Sequences

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