[math-fun] RE: The "commas" sequence revisited

10 Dec 2007

      Hello David and Math-Fun,

thanks a lot for your computations, David -- I'll update my
webpages asap -- you gave me the idea of a puzzle:

Let a and b be two integers linked by an hyphen in such way
that [the number formed by the last digit of a and the first 
digit of b] is the absolute difference between a and b. Example:

  a - b
 28 - 109

(28 + 81 = 109)

We admit zero as last digit of a : "30 - 28" is ok.

We always read the hyphanated couples "a - b" from left to right.

Now look at the integer 22:

    20        1
       \    /
         22
       /    \
    64        46

The hyphens are set diagonally, but you can still check that:

20 + 02 = 22
1 + 21 = 22
42 + 22 = 64
22 + 24 = 46

Another example with 32 in the center:

              1
            /
    85 - 32 - 11
            \
              46

Question: find the smallest (unique?) integer X fitting this diagram:

     a       b
       \   /
         X - c
       /   \
     d       e

More informations here (but no spoiler):
http://www.cetteadressecomportecinquantesignes.com/Commas.htm

Best,
É.

-----Message d'origine-----
De : David Wilson [mailto:davidwwilson@comcast.net] 
Envoyé : dimanche 9 décembre 2007 17:59
À : Eric Angelini; math-fun
Objet : Re: The "commas" sequence revisited

A commas sequence starting with any n > 0 will eventually include an element 
< 100. I'm sure I could prove this, it would be a very messy and distasteful 
proof.

Once the sequence reaches a value < 100, it must shortly terminate at one of 
the values 18, 27, 36 or 45, or else enter the loop (6, 73, 104, 59, 150, 
149, 54, 7, 85, 32)

I doubt there is a simple way to determine the fate of a sequence starting 
with n.

The starting number 1 leads to the terminating sequence

    1, 12, 35, 94, 135, 78, 159, 63, 30, 28, 109, 18

----- Original Message ----- 
From: "Eric Angelini" <Eric.Angelini@kntv.be>
To: "math-fun" <math-fun@mailman.xmission.com>; "Sequence Fans" 
<seqfan@ext.jussieu.fr>
Sent: Thursday, December 06, 2007 12:31 PM
Subject: The "commas" sequence revisited
...
Hello Math-Fun and SeqFans,
I would like to come back to this, if you don't mind
(this was one year ago):
http://www.groupsrv.com/hobby/about405191.html
Ed Murphy had asked (quote):
-----
...
What if d = abs(a(n+1)+a(n)), i.e. the sequence is not
necessarily monotonically increasing? Which choices
for a(1) result in a block? Which choices result in a loop?
Sample sequence:
1, 12, 35, 94, 135, 78, 159, 251, 239, 148, 62, 91, 74,
115, 166, 105, 156, 88, 4, 48, 129, 221, 209, 118, 199,...
-----
[Me]:
-- Could someone check if the sequence starting with a(1)=1
ever enters a loop (should we always keep the terms as low
as possible -- although we would still avoid blocks)?
-- Is there a seq which could be a loop by itself?
  a-->b-->c-->d-->e-->f-->a
Best,
É.
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