The Simpsons opening sequence enter this cartoon universe of "The Simpsons". Finally, after a couple of decades, they've gotten closer to what I had in my mind. Not perfect, but better." The two original variations were further expanded to these variations: The billboard gag is a running visual joke added to the opening sequence with the updated 2009 high-definition opening. In the gag, a billboard is seen on the roof of the building across the street from the elementary school as the camera pans through the town. The billboard changes every episode. The first episode with a billboard gag was "Take My Life, Please"
Automatic sequence is a fixed point of φ—that is, if "w" = φ("w")—then "s" = τ("w") is a "k"-automatic sequence. This result is due to Cobham, and it is referred to in the literature as "Cobham's theorem". Conversely, every "k"-automatic sequence is obtainable in this way. Let "k" ≥ 2. The "k-kernel" of the sequence "s"("n") is the set of subsequences In most cases, the "k"-kernel of a sequence is infinite. However, if the "k"-kernel is finite, then the sequence "s"("n") is "k"-automatic. It follows that a "k"-automatic sequence is necessarily a sequence on a finite alphabet. Let "u"("n") be a sequence