# [seqfan] Three triangles

Eric Angelini Eric.Angelini at kntv.be
Wed May 8 11:05:00 CEST 2013

```Hello SeqFans,
are these three pyramids worth an entry in the OEIS?

1
2    3
4    6    9
10   14   20   29
30   40   54   74   103
104  134  174  228  302   405
406   ...

Each term is the absolute difference between the two terms
immediately below.
When you read from left to right, and from top to bottom,
the sequence of integers is monotonically increasing (by
construction):

1,2,3,4,6,9,10,14,20,29,30,40,54,74,103,104,134,174,228,...

This seq doesn't seem to be in the OEIS -- or is it in
another form?

The leftmost diagonal "/" echoes of course the rightmost
one "\" -- as we start always a new row from the left with
the last integer of the row before, increased by 1 (see the
example above with 405 and 406).
-----------------------------------------------------------

Absolute differences might be used in another way:

1
2    3
4    6    9
7   11   17   26
5   12   23   40    66
8   13   25   48   88   154
10  18   31   56   104  192   346
...

We see here that each row is monotonically increasing and
always starting with the smallest unused integer so far
completing the row).
------------------------------------------------------------

The 3rd "absolute differences" pyramid wants to be a permutation
of the natural numbers: is it possible to build such a triangle?
(The process will need a lot of backtracking, I'm afraid...):

1
2    3
4    6    9
7   11    5   14
8   15   26   21    35
10   18   33   59   38   73
12  22   40   73 ...

... ouch! We have already used 73 (at the end of the previous row)