The Simpsons opening sequence couch gag is shown, before the credits are displayed on a new widescreen flat panel television. Sometimes, the flat panel television falls to the floor and breaks, but unlike "Bart Gets a Z", the episode will begin then anyway. In the 500th episode, the opening sequence was a montage of all previous couch gags. Creator Matt Groening developed a lengthy opening sequence for the first season of "The Simpsons", in order to cut down on the animation necessary for each episode, but devised the two gags as compensation for the repeated material each week. In the first of the original
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