[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
not leading to the repetition of an already used term (in
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)
This means that the last row cannot start with 12...
If all this is old hat, please forgive.
Best,
É.
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