Truncated Ulam sequences and the question of eventual periodicity

Whether the Ulam sequence 1, 2, 3, 4, 6, 8, 11, … is eventually periodic is open (Erdős problem #342). We collect some elementary statements and numerical observations about periodic approximations of it. If only the first K terms may serve as small summands, the resulting sequence C_K is eventually periodic, by Finch's argument. Numerically its frequency approaches Steinerberger's frequency β = α/2π as K grows (to within about 2·10^−4 for 30 < K ≤ 80 and about 10^−6 for 192 ≤ K ≤ 1024, where periods were found for all K ≤ 80 and for K = 96, 128, 192, 256, 768), while for K > 30 the rational frequencies are not good approximations of β in the sense of continued fractions. If β is irrational, periodic sets are equidistributed along the rotation by β, so a persistent angular bias of the Ulam sequence rules out periodicity; for the truncations this means that two limits cannot be interchanged. Conversely, eventual periodicity would force the set of small summands to be finite; below 2^27 there are 299 of them, and new ones keep appearing. Forbidding new outliers (in Gibbs's sense) beyond some point gives sequences that, if β is irrational, are never eventually periodic provided they are infinite. Some of them are nevertheless exactly ordered: with the first six outliers frozen, the sequence coincides from 755 on with the integers whose angle lies in an explicit union of 17 intervals, a model set (a computer-assisted proof, valid for all irrational β in an interval containing α/2π). Others lock onto near-periods of the rotation for long stretches, and with more frozen outliers this order is no longer seen in the ranges we tested. The Ulam sequence itself shows no excess of coincidences at the denominators of the convergents of β. Nothing is proved about the Ulam sequence itself without hypotheses. The proofs are short and written out in full, but their validity is left open to external verification; see the note at the end of the paper. The accompanying archive contains the programs and the recorded outputs.