The Simpsons opening sequence in the first episode to use the new opening sequence, "Take My Life, Please", where the family chases their couch on a tour across the world. Another long couch gag was in the show's 500th episode "At Long Last Leave," showing a montage of previous couch gags. In 2006, the British television channel Sky 1 began advertising "The Simpsons" using a live-action recreation of the series' opening sequence directed by Chris Palmer. Except for the very first shot in which the logo appears out of the clouds, every piece of the opening is present in this version, with even multiple
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