The Simpsons opening sequence had been introduced during the previous season. Starting with this season, there were three versions of the opening: a full roughly 1 minute 15 second long version, a 45 second version and a 25 second version. This gave the show's editors more leeway. "Take My Life, Please" (season 20) was the first episode of "The Simpsons" to air in 720p high-definition television, though not the first time "The Simpsons" appeared in high-definition, as "The Simpsons Movie" was rendered in HD. This episode was the first to feature the new opening sequence. It was the first major permanent change to the
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