The Simpsons opening sequence of the opening titles appeared in "White Christmas Blues". A completely different sequence was created for "The Simpsons Movie" and features an orchestrated version of "The Simpsons Theme" as adapted by Hans Zimmer. The cumulus clouds are displayed in 16:9 television aspect ratio, with black matte bars at either end of the screen. As "The Simpsons" logo appears out of the clouds, Professor Frink flies past in one of his inventions carrying a banner marked "MOVIE" and proclaims "Moo-vie! On the big screen!" (On the movie's DVD menu he says, "On the small screen!" when the menu appears, but then
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