This constant T = 1.83928... (A058265) is the real root of X^3-X^2-X-1. Its Beatty sequence (or spectrum) is the sequence a(n) = floor(nT), A158919. I found a lovely (conjectured) recurrence for a related sequence, but as a sanity check I applied the same analysis to A158919 itself. What I found is depressing because A158919 is regular for 20000 terms but then a new pattern appears. This sort of thing is common in Pisot sequences (e.g. A007699), but I was suprised to encounter it here. It raises the possibility that my lovely conjecture may be only a mirage. Beatty sequences have been studied since the 1920s, especially in Canada, and there are dozens of paperss. Could someone explain what is going on and point me to the right reference? Start with the Beatty sequence and apply some operations: A158919, floor(nT): 0, 1, 3, 5, 7, 9, 11, 12, ... [I used 100000 terms] DIFF: 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, [DIFF = "differences"] RUNS: 1, 5, 1, 5, 1, 5, 1, 5, 1, 6, 1, 5,... [RUNS = "run lengths"] BISECT: 5, 5, 5, 5, 6, 5, 5, 5, 6, 5, 5, 5, 5, 6, 5,.. [take alternate terms] RUNS: 4, 1, 3, 1, 4, 1, 3, 1, 4, 1, 3, 1, 4, 1, 3... [now down to 7144 terms] BISECT: 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3,... [3467 terms] RUNS: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... EXCEPT FOR 2's at terms 612, 1224, 1836, 2448, 3060. Help! Presumably the trouble at term 612 comes from the 305 in the continued fraction for T, (A019712): 1, 1, 5, 4, 2, 305, 1, 8, 2, 1,... ?