[seqfan] Filling squares and triangles (with digits)

[Eric Angelini]

Hello SeqFans,
It is possible to write down in a 5 x 5
square (25 cells)  the integers from 1
to 17 -- if you use one digit per cell:

1 2 3 4 5 
6 7 8 9 1
0 1 1 1 2
1 3 1 4 1
5 1 6 1 7

The 4 x 4 square is impossible to fill
exactly, using the same constraint:

1 2 3 4
5 6 7 8
9 1 0 1
1 1 2 1 3

We see that 12 leaves an empty cell,
and 13 needs one too much.

What are the exact "square-filling" integers?

I guess S starts:
S=1,4,9,17,29,45,65,89,111,144,183,228,...

The equivalent seq T could be constructed
for exact "right-triangle-filling" integers:

T=1,3,6,12,15,27,32,50,57,...

None of those are in the
O
N L
I N E
E N C Y
C L O P E
D I A O F I
N T E G E R S
E Q U E N C E S
Best,
É.
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(What about in other bases?)