[seqfan] Re: Triplets and pair sums

[Frank Adams-Watters]

I think any such iteration that doesn't lead to a loop is going to have values that grow exponentially. So I would not expect every number to occur.

This is pretty easy to compute in a spreadsheet. You can sort a triplet (into new locations) by taking the min for the first value, max for the third,  and sum minus the other two computed values for the second.

Franklin T. Adams-Watters

-----Original Message-----
From: Eric Angelini <Eric.Angelini at kntv.be>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Cc: alexandre.wajnberg <alexandre.wajnberg at skynet.be>
Sent: Thu, Aug 27, 2015 11:00 am
Subject: [seqfan] Triplets and pair sums



Hello SeqFans,
(A) Order the triplet [a, b, c] such that a <= b <= c
   -> for
example [1, 2, 3,]
(B) Change the sign of the "middle value"
   -> here: [1, -2,
3]
(C) Compute the new triplet [a+b, a+c, b+c]
   -> here: [-1, 4, 1]
(D) Go
back to (A) and iterate.

We'll have, from the start :
(the 4th column will be
discussed later)

  --start-
 [1, 2, 3,]

 --sign--       ---new---       --- re
---      4th column
 _change_       _triplet_       __order___     
(a+b+c)/2

[1, -2, 3]  -> [-1, 4, 1] ->   [-1, 1, 4]          2
[-1, -1, 4] ->
[-2, 3, 3] ->   same                2
[-2, -3, 3] -> [-5, 1, 0] ->   [-5, 0, 1] 
-2
[-5, 0, 1]  -> [-5, -4, 1] ->  same               -4
[-5, +4, 1] -> [-1, -4,
5] ->  [-4, -1, 5]         0
[-4, +1, 5] -> [-3, 1, 6] ->   same               
2
[-3, -1, 6] -> [-4, 3, 5] ->   same                2
[-4, -3, 5] -> [-7, 1, 2]
->   same               -2
[-7, -1, 2] -> [-8, -5, 1] ->  same              
-6
[-8, +5, 1] -> [-3, -7, 6] ->  [-7, -3, 6]        -2
[-7, +3, 6] -> [-4, -1,
9] ->  same                2
[-4, +1, 9] -> [-3, 5, 10] ->  same               
6
[-3, -5, 10] > [-8, -7, 5] ->  same                5
[-8, +7, 5] -> [-1, -3,
12] -> [-3, -1, 12]        4
[-3, +1, 12] > [-2, 9, 13] ->  same              
10
[-2, -9, 13] > [-11, 11, 4] -> [-11, 4, 11]        2
[-11, 4, 11] > [-7, 0,
15] ->  same                4
[-7, 0, 15] -> [-7, 8, 15] ->  same              
12
[-7, 8, 15] -> [-1, 8, 23] ->  same               15
[-1, -8, 23] > [-9, 22,
15] -> [-9, 15, 22]       14
[-9, -15, 22]  [-24, 13, 7] -> [-24, 7, 13]      
-4
[-24, -7, 13]  [-31, -21, 6] > same              -23
[-31, +21, 6]  [-10,
-25, 27]  [-25, -10, 27]     -4
[-25, +10, 27] [-15, 2, 37] -> same             
12
[-15, -2, 37]  [-17, 22, 35] > same               20
etc.

If, instead of [1,
2, 3] we'd decided to start with
the triplet [-1, 0, 1], we'd be blocked - as
all
triplets of the form [-a, 0, a] are fixed points.

I'd very much like to see
the "4th column" graph of
the starting triplet [-1, 0, 2]...

And to know if
there are more fixed points (or loops).
And to learn if all (positive and
negative) integers
will appear at some point, in the "new triplet" column
that
starts with [-1, 0, 2]...

Best,
É.


    





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