[seqfan] Triplets and pair sums

[Eric Angelini]

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.