A note on Buell’s Theorem on length four Büchi sequences

Büchi sequences are sequences whose second difference of squares is the sequence (2,..., 2), like for instance (6, 23, 32, 39) – so they can be seen as a generalization of arithmetic progressions. No (non-trivial) length 5 Büchi sequence is known to exist. Length four Büchi sequences were parameterized by D. A. Buell in 1987. We revisit his theorem, fixing the statement (about 26% of the Büchi sequences from R. G. E. Pinch's 1993 table were missed), and giving a much simpler proof.