N0. Hidden structure It has been observed that the first 10 million Ulam numbers satisfy except for the four elements (this has now been verified for the first Ulam numbers). Inequalities of this type are usually true for sequences exhibiting some form of periodicity but the Ulam sequence does not seem to be periodic and the phenomenon is not understood. It can be exploited to do a fast computation of the Ulam sequence (see External links). Generalizations The idea can be generalized as (u, v)-Ulam numbers by selecting different starting values (u, v). A sequence of (u, v)-Ulam numbers is regular if the sequence of differences between consecutive numbers in the sequence is eventually periodic. When v is an odd number greater than three, the (2, v)-Ulam numbers are regular. When v is congruent to 1 (mod 4) and at least five, the (4, v)-Ulam numbers are again regular. However, the Ulam numbers themselves do not appear to be regular. A sequence of numbers is said to be s-additive if each number in the sequence, after the initial 2s terms of the sequence, has exactly s representations as a sum of two previous numbers. Thus, the Ulam numbers and the (u, v)-Ulam numbers are 1-additive sequences. If a sequence is formed by appending the largest number with a unique representation as a sum of two earlier numbers, instead of appending the smallest uniquely representable number, then the resulting sequence is the sequence of Fibonacci numbers. Notes References External links Ulam Sequence from MathWorld Fast computation of the Ulam sequence by Philip Gibbs Description of Algorithm by Donald Knuth The github page of Daniel Ross Integer sequences