This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
If a(2) were 1, a(2) would have to be 2. a(2) cannot be both 1 and 2, so a(2) cannot be 1. If a(2) were 2, the next 2 after a(1) would appear at a(3). The next 2 actually appears at a(2), so a(2) cannot be 2. If a(2) were 3, a(4) would have to be 2. If a(4) were 2, a(3) and a(5) would have to have the same value. If a(3) and a(5) were equal, the pattern X, 3, Y, 2 would appear later in the sequence. The pattern X, 3, Y, 2 is invalid because X would have to be 2. X cannot be 2 because it comes after the 2 at a(1) and before the 2 that succeeds Y. This behavior is inconsistent with the definition of the sequence. The next possible value for a(2) is 4, which causes no trivial inconsistencies (as with a(2)=1 or a(2)=2) or logical inconsistencies (as with a(2)=3). The n-th term of the sequence is the lowest value that does not cause inconsistencies of either sort.
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