The Period(w) command computes the period of a string (word) w. A positive integer p is said to be a period of a string s if, for all integers i with $1<=i<=\mathrm{length}\left(s\right)-p$, one has $s_i=s_{i+p}$. The least period of the string s is called the period of s.

All of the StringTools package commands treat strings as (null-terminated) sequences of $8$-bit (ASCII) characters.  Thus, there is no support for multibyte character encodings, such as unicode encodings.

$\mathrm{with}\left(\mathrm{StringTools}\right)\&colon;$

$\mathrm{Period}\left(''aaa''\right)$

$1$

$\mathrm{Period}\left(''abcde''\right)$

$5$

$\mathrm{Period}\left(''abcabc''\right)$

$3$

$\mathrm{Period}\left(''abcabcab''\right)$

$3$

$\mathrm{seq}\left(\mathrm{Period}\left(\mathrm{Fibonacci}\left(n\right)\right)\&comma;n=1..10\right)$

$1,2,2,3,5,8,13,21,34,55$

$\mathrm{seq}\left(\mathrm{combinat}\left['\mathrm{fibonacci}'\right]\left(n\right)\&comma;n=1..10\right)$

$1,1,2,3,5,8,13,21,34,55$

$\mathrm{seq}\left(\mathrm{Period}\left(\mathrm{ThueMorse}\left(n\right)\right)\&comma;n=1..10\right)$

$1,2,3,3,3,5,6,6,6,6$

$\mathrm{`Periodic?`}≔w\mapsto \mathrm{evalb}\left(\mathrm{Period}\left(w\right)\le \mathrm{floor}\left(\frac{\mathrm{length}\left(w\right)}{2}\right)\right)$

$\mathrm{Periodic?}≔w\mapsto \mathrm{evalb}\left(\mathrm{StringTools}:-\mathrm{Period}\left(w\right)\le ⌊\frac{\mathrm{length}\left(w\right)}{2}⌋\right)$

$\left(\mathrm{Period}\&comma;\mathrm{length}\&comma;\mathrm{`Periodic?`}\right)\left(''ababababababababa''\right)$

$2,17,\mathrm{true}$

$\left(\mathrm{Period}\&comma;\mathrm{length}\&comma;\mathrm{`Periodic?`}\right)\left(''abcdeab''\right)$

$5,7,\mathrm{false}$