Can you give me a reference to the definition and significance of the Ulam sequence?
Some information can be found at http://mathworld.wolfram.com/UlamSequence.html Here's an abstract of a paper in J Comb Th (A). I'll leave it to someone else to explain the significance. 95h:11010 Schmerl, James(1-CT); Spiegel, Eugene(1-CT) The regularity of some $1$-additive sequences. J. Combin. Theory Ser. A 66 (1994), no. 1, 172--175. 11B13 (11B39 11B83) -------------------------------------------------------------------------------- Given two integers $u,v$, $0<u<v$, S. M. Ulam \ref[cf. R. K. Guy, problems in number theory, Springer, New York, 1981; MR 83k:10002 (Problem C4)] defined a strictly increasing sequence $(u_n)_{n\geq 1}$ of integers, with base $(u,v)$, by $u_1=u$, $u_2=v$ and, for all $n\geq 1$, $u_{n+2}$ to be the least integer which exceeds $u_{n+1}$ and has a unique representation $u_{n+2}=u_i+u_j$, where $1\leq i<j\leq n+1$. Sequences with base $(1,v)$ present erratic behavior, while the sequence with base $(2,5)$ is "regular", which means that the sequence of differences $d_n=u_{n+1}-u_n$ is ultimately periodic; in the (2,5) case, $d_{n+32}=d_n$ for all $n\geq 7$. This fact was first noticed by the author of Zazie dans le M\'etro \ref[R. Queneau, J. Combin. Theory Ser. A 12 (1972), 31--71; MR 46 #1741]. For further results and references see two papers of S. R. Finch\ \ref[Fibonacci Quart. 29 (1991), no. 3, 209--214; MR 92j:11009; J. Combin. Theory Ser. A 60 (1992), no. 1, 123--130; MR 93c:11009]. In the present note, the authors prove a conjecture of Finch, stating that, if $v>3$ is an odd integer, then the sequence with base $(2,v)$ has only two even terms: $u_1=2$ and $u_{(v+7)/2}=2v+2$. The proof is elementary and self-contained. Regularity follows because Finch proved, among other things, that if an Ulam sequence has finitely many even terms, then it is regular. --------------------------------------------------------------------- This is from MathSciNet. If you go to this abstract in MathSciNet the links to the papers mentioned in the abstract will be live.