At 06:55 PM 2/14/2003, Klaus Brockhaus wrote:

Using David Wilson's idea and Jud McCranie's data, I made some 216x246 gif-files (attached) which illustrate the observed phenomenon.

I don't know if this is known (or interesting or significant), but there are many "pseudo-repetitions" in the Ulam sequence. By this I mean a relatively large number, n, so that U(s+i+p)=(s+i)+d for i=0 .. n-1, for some constants s, p, and d. For instance n=134 d=4131173 The 134 Ulam numbers starting with 51497 (Ulam number, not index) and the 134 Ulam numbers starting at 4182670 each differ by 4131173 (the corresponding terms, that is).

At 09:42 PM 2/14/03 -0500, you wrote: I'm wondering whether these repetitions gradually 'accrete', that is, get incorporated in larger and larger repeated subsequences. I'm also wondering whether the repetition is due to an innate tendency toward those particular subsequences, or whether there is some sort of copying process that tends to Xerox earlier subsequences under some circumstances. If the former, we would expect to see some of the same subsequences in the first differences of Ulam(1,3); if the latter, we would expect different repeated subsequences in Ulam(1,3). You see, if the content of the repeated subsequences is partly arbitrary, that is, if, in the right circumstances, anything could be copied, then we may have actual natural selection happening as the different sequences compete for space and 'attention' by the copying mechanism. I await developments. -A

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.

Don't miss the amazing solution to misere Kayles by the amateur William L Sibert that's been added at the end of the new volume two (what used to be the old Chapter 13 in Vol I the first edition of Winning Ways). Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/home.htm ----- Original Message ----- From: "Ed Pegg Jr" <ed@mathpuzzle.com> To: <math-fun@mailman.xmission.com> Sent: Thursday, March 13, 2003 5:27 PM Subject: [math-fun] Winning Ways, 2nd ed, Vol 2

Is now available. http://www.akpeters.com/book.asp?bID=149

Which leads me to suppose that one of the following is true: 1. Our esteemed authors are too modest to mention it. 2. When a book finally hits the shelves, the authors are as surprised as anyone else. 3. They were planning a big post about it

Vol 3 and vol 4 are hypothetically being released in May.

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Don't miss the amazing solution to misere Kayles by the amateur William L Sibert that's been added at the end of the new volume two (what used to be the old Chapter 13 in Vol I the first edition of Winning Ways). Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/home.htm ----- Original Message ----- From: "Ed Pegg Jr" <ed@mathpuzzle.com> To: <math-fun@mailman.xmission.com> Sent: Thursday, March 13, 2003 5:27 PM Subject: [math-fun] Winning Ways, 2nd ed, Vol 2

Is now available. http://www.akpeters.com/book.asp?bID=149

Which leads me to suppose that one of the following is true: 1. Our esteemed authors are too modest to mention it. 2. When a book finally hits the shelves, the authors are as surprised as anyone else. 3. They were planning a big post about it

Vol 3 and vol 4 are hypothetically being released in May.

_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun