Re: [math-fun] Ulam(1,2)
"David Wilson" <davidwwilson@attbi.com> wrote:
I took a look at Ulam(1,2), the Ulam sequence starting with (1, 2) and including every subsequent number which is a unique sum of distinct earlier terms. This is Sloane's A002858.
... Can anyone confirm my data?
Did you intend to use zero-based indexing? If u(0)=1, u(1)=2, I concur that u(1000)=12336, u(1999)=25511. Otherwise one of us has a bug. I also get u(10000)=132790, u(50000)=676043, u(74084)=1000002. But there's a pretty definite bug in the histogram. My histogram of f(n)=(u(n) mod 21.6)/21.6 agrees with yours in its general trend, but differs in particulars. For f(1000...1999) in 20 buckets I get 113 101 119 98 87 70 58 40 31 8 5 0 0 2 4 14 22 64 66 98 rather than your 110 101 120 97 90 67 58 40 31 9 4 0 0 2 7 11 22 64 66 101. As a test, I find that the five numbers in the [.5,.55) bucket arise from u(1091)=13533 (f=0.527771) u(1210)=14894 (f=0.53704834) u(1730)=21870 (f=0.5) u(1748)=22151 (f=0.50927734) u(1947)=24851 (f=0.50927734). What four numbers do you have? I agree exactly with Klaus Brockhaus's data for f(1000...1259), so I think I'm slightly likelier to be correct, but only slightly, and I'm pretty suspicious about the zero-based indexing. My data suggests that 21.6 should be about 21.60157 . Dan
There are three differences between my computations and yours. (1) I use Perl (2) I index from zero and (3) I wrote the algorithm myself. Any of these may account for discrepancies between our results, and since you and Brockhaus agree, I will let you be right, it isn't worth it to me to find my error. All I was interested in was whether the observed bunching phenomenon was real, or whether I was totally off the wall. It appears that both you and Brockhaus have confirmed the bunching with period about 21.6. Do you think this is a constant? If so, what would cause this regular bunching? If you plotted the integers 216 per row and colored the Ulams black and the others white, I bet you would get a pretty set of stripes. Anyway, thanks for confirming my observation. ----- Original Message ----- From: "Dan Hoey" <Hoey@aic.nrl.navy.mil> To: "David Wilson" <davidwwilson@attbi.com> Cc: "Math Fun" <math-fun@mailman.xmission.com>; "Klaus Brockhaus" <klaus-brockhaus@t-online.de> Sent: Monday, February 10, 2003 6:07 PM Subject: Re: [math-fun] Ulam(1,2)
"David Wilson" <davidwwilson@attbi.com> wrote:
I took a look at Ulam(1,2), the Ulam sequence starting with (1, 2) and including every subsequent number which is a unique sum of distinct earlier terms. This is Sloane's A002858.
... Can anyone confirm my data?
Did you intend to use zero-based indexing? If u(0)=1, u(1)=2, I concur that u(1000)=12336, u(1999)=25511. Otherwise one of us has a bug. I also get u(10000)=132790, u(50000)=676043, u(74084)=1000002.
But there's a pretty definite bug in the histogram. My histogram of f(n)=(u(n) mod 21.6)/21.6 agrees with yours in its general trend, but differs in particulars. For f(1000...1999) in 20 buckets I get
113 101 119 98 87 70 58 40 31 8 5 0 0 2 4 14 22 64 66 98
rather than your
110 101 120 97 90 67 58 40 31 9 4 0 0 2 7 11 22 64 66 101.
As a test, I find that the five numbers in the [.5,.55) bucket arise from u(1091)=13533 (f=0.527771) u(1210)=14894 (f=0.53704834) u(1730)=21870 (f=0.5) u(1748)=22151 (f=0.50927734) u(1947)=24851 (f=0.50927734). What four numbers do you have?
I agree exactly with Klaus Brockhaus's data for f(1000...1259), so I think I'm slightly likelier to be correct, but only slightly, and I'm pretty suspicious about the zero-based indexing.
My data suggests that 21.6 should be about 21.60157 .
Dan
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Using David Wilson's idea and Jud McCranie's data, I made some 216x246 gif-files (attached) which illustrate the observed phenomenon. ulampic0.gif: numbers from 1 to 53136; ulampic1.gif: numbers from 1000001 to 1053136; ulampic2.gif: numbers from 2000001 to 2053136; ulampic3.gif: numbers from 3000001 to 3053136; ulampic4.gif: numbers from 4000001 to 4053136; Klaus ------------------------------------------------------------------- David Wilson wrote: [......]
It appears that both you and Brockhaus have confirmed the bunching with period about 21.6. Do you think this is a constant? If so, what would cause this regular bunching? If you plotted the integers 216 per row and colored the Ulams black and the others white, I bet you would get a pretty set of stripes. [......]
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).
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).
I've now found such a set of length 251. To state it in another way, the first differences of these two sections of length 251 are the same.
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?
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.
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.
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
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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participants (9)
-
Allan C. Wechsler -
Dan Hoey -
David Wilson -
Ed Pegg Jr -
Edwin Clark -
John McCarthy -
Jud McCranie -
klaus-brockhaus@t-online.de -
Thane Plambeck