OFFSET
2,1
COMMENTS
See A249866 for comments and references.
For the sorted areas of all primitive Pythagorean triangles (x, y, z) with, say y even, see A024406.
Note that in a row > N there may appear smaller numbers than the maximal number up to row N. Therefore the sorted nonvanishing numbers up to a given row N will in general not produce a subsequence of A024406. The minimal areas in rows n = 2..20 are 6, 30, 60, 180, 210, 546, 504, 1224, 990, 2310, 1716, 3900, 2730, 6090, 4080, 8976, 5814, 12654, 7980. For example, one has to go up to row n = 16 to cover all areas <= 4080.
See the link for more details on a safe row number n = N to cover all areas not exceeding a given one, and also for all areas <= 10^6 with their squarefree parts. - Wolfdieter Lang, Nov 25 2016
LINKS
Wolfdieter Lang, First rows of the triangle.
FORMULA
T(n, m) = n*m*(n+m)(n-m) if n > m >= 1, (-1)^(n+m) = -1 and gcd(n,m) = 1, else 0.
EXAMPLE
The triangle T(n, m) begins:
n\m 1 2 3 4 5 6 7 8 9 10 11
2: 6
3: 0 30
4: 60 0 84
5: 0 210 0 180
6: 210 0 0 0 330
7: 0 630 0 924 0 546
8: 504 0 1320 0 1560 0 840
9: 0 1386 0 2340 0 0 0 1224
10: 990 0 2730 0 0 0 3570 0 1710
11: 0 2574 0 4620 0 5610 0 5016 0 2310
12: 1716 0 0 0 7140 0 7980 0 0 0 3036
...
For more rows see the link.
T(5, 2) = 210 for the primitive triangle (21, 20, 29).
T(6, 1) = 210 for the primitive triangle (35, 12, 37).
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Dec 03 2014
STATUS
approved