Eric Angelini Eric.Angelini at kntv.be
Sat May 16 15:56:37 CEST 2015

```Hello SeqFans,
Here is a way to build an infinite 0-additive array, I guess.

Always try to fill the leftmost empty cell of the top row with
the smallest unused integer "a" so far.

If this is not possible, try to fill the leftmost empty cell of
the next row with "a".

If this is not possible, try to fill the leftmost empty cell of
the next row with "a" (etc.)

Reminder: 0-additivity means that no "a" is the sum of any two
integers belonging to the row where "a" stands.

I get (by hand) with this method, for the first 50 positive
numbers:

+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| 1| 2| 4| 7|10|13|16|19|22|25|28|31|34|37|40|43|46|49|..|..|
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| 3| 5| 6|12|14|21|23|30|32|39|41|48|..|..|  |  |  |  |  |  |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| 8| 9|11|15|18|35|36|38|42|..|..|  |  |  |  |  |  |  |  |  |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
|17|20|24|26|27|29|33|45|..|..|  |  |  |  |  |  |  |  |  |  |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
|44|47|..|..|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+

Example:
47 is not in the 1st row because 1 and 46 sum to 47
47 is not in the 2nd row because 6 and 41 sum to 47
47 is not in the 3rd row because of 11 and 36
47 is not in the 4th row because of 20 and 27
... thus 47 occupies the leftmost empty square of the 5th row.

The first row is https://oeis.org/A033627  ("0-additive sequence:
not the sum of any previous pair.")
The pattern is obvious: "2 together with numbers of form 3k+1"

The second row is not in the OEIS -- and the pattern seems to be
"3,5,6,12 together with +2,+7,+2,+7,+2,+7,..."

There arn't enough terms in the 3rd row to read a pattern, I think.

Would it be of interest to enter this array by anti-diagonals?

S = 1,2,3,4,5,8,7,6,9,17,10,12,11,20,44,... (not in the OEIS)

Question #1:
Will we see, in this infinite array, a 0-additivity in the columns too?
So far yes, but I have my doubts...

Then comes the question #2:
If we fix the size of an array, is it always possible to fill the said array

I've done this quickly by hand, for [m x m] arrays (m>=3, of course);
all integers from 1 to m^2 must be present:

m=3:

1  2  4
3  5  6
7  8  9

m=4:

1  2  4  7
3  5  6 10
8  9 11 12
13 15 14 16

m=5

1  2  4  7 10
3  5  6 12 13
8  9 11 14 15
16 17 18 20 19
21 23 25 22 24

m=6:

1  2  4  7 10 13
3  5  6 12 14 16
8  9 11 15 18 21
17 19 20 23 22 24
26 25 27 28 29 30
31 32 34 33 35 36

Question #3:
For a given m, how many different 0-additive squares are possible?

Question #4:
Are some [m x n] shapes (with m and n >=3) impossible to fill with
consecutive integers (starting with "1")?

I couln't insert 17 here (m=3, n=6)

1  2  4  7 10 13
3  5  6 12 14 15
8  9 11 16 18 __

[Sorry for the long post, Gérard -- especially if this is old hat.
I'll keep quite for the w.-e. anyway!-]
[Gérard is the list-admin, as you all know]
Best,
É.

```