[seqfan] A triangular observation (nothing proved)
Jonas Wallgren
jwc at ida.liu.se
Thu Mar 4 13:29:20 CET 2010
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
More information about the SeqFan
mailing list