On the number of words containing the factor image

In this paper a recurrence relation satisfied by the number image of words of length n over an alphabet A of cardinality m image not containing the factor image image is deduced. Let image be a sequence of positive integers. From [I. Tomescu, A threshold property concerning words containing all short factors, Bull. EATCS 64 (1998) 166–170] it follows that if image then almost all words of length n over A contain the factor image as image. Using the properties of the roots of the recurrence satisfied by image it is shown that if image then this property is false. Moreover, if image then image, where image denotes the set of words of length n over A containing the factor image.