[seqfan] A triangular observation (nothing proved)

[Jonas Wallgren]

Hi!

This is (the top of) a triangle for r=3.
The left edge is n, the right edge is n². (n is the row number.)

            0
          1   1
        2   0   4
      3   0   0   9
     4  2   2   2  16
    5 2   6   6   4  25
  6  3  3  12   9   9  36
 7 6  8  10  22  20  18  49
8 7 13 19  18  41  35  29  64

Every element inside the triangle is the sum of r+1 elements r rows up. E.g.
18 in the middle of the bottom row = 2+6+6+4 from 3 rows up. The elements
summed are those placed symmetrically around a vertical line through the sum
and as close to it as possible.
All elements outside the triangle = 0.

The sum of all elements in each row gives the following sequence, together
with its 1st and 2nd differences:

0 2 6 12  26  48  78  140  234   360   614   996   1506    2528    4062
 2 4 6  14  22  30  62   94   126   254   382   510    1022    1534
  2 2  8   8   8  32   32   32   128   128   128    512     512

For an arbitrary r the 2nd difference (seems to) consists of:
r-1 copies of 2,
  r copies of 2*(r+1),
  r copies of 2*(r+1)²,
  r copies of 2*(r+1)³,
  ...

Maybe this is interesting in some way ... ?

								/Jonas Wallgren