For n > 2, any three consecutive Ulam numbers (Un−1, Un, Un+1) as integer sides will form a triangle.
In any sequence of 5 consecutive positive integers {i, i + 1,..., i + 4}, i > 4 there can be a maximum of 2 Ulam numbers.
[7] Ulam numbers are pseudo-random and too irregular to have tight bounds.
Nevertheless from the properties above, namely, at worst the next Ulam number Un+1 ≤ Un + Un−2 and in any five consecutive positive integers at most two can be Ulam numbers, it can be stated that where Nn are the numbers in Narayana’s cows sequence: 1,1,1,2,3,4,6,9,13,19,... with the recurrence relation Nn = Nn−1 +Nn−3 that starts at N0.
It has been observed[8] that the first 10 million Ulam numbers satisfy
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).
The idea can be generalized as (u, v)-Ulam numbers by selecting different starting values (u, v).