A249869 Triangle giving the area of primitive Pythagorean triangles, with zero entries for non-primitive triangles.

6, 0, 30, 60, 0, 84, 0, 210, 0, 180, 210, 0, 0, 0, 330, 0, 630, 0, 924, 0, 546, 504, 0, 1320, 0, 1560, 0, 840, 0, 1386, 0, 2340, 0, 0, 0, 1224, 990, 0, 2730, 0, 0, 0, 3570, 0, 1710, 0, 2574, 0, 4620, 0, 5610, 0, 5016, 0, 2310, 1716, 0, 0, 0, 7140, 0, 7980, 0, 0, 0, 3036

Offset: 2

Author

Wolfdieter Lang, Dec 03 2014

Keywords

• nonn
• easy
• tabl

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

Examples

			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).
		

Links

• Wolfdieter Lang, First rows of the triangle.
• Wolfdieter Lang, A Note on the Area table A249869 for Primitive Pythagorean Triangles.

Crossrefs

Cf. A024406, A249866, A258150 (one sixth of this triangle), A225949 (leg sums), A225951 (perimeters), A222946 (hypotenuses), A208854 (odd catheti), A208855 (even catheti), A278711.

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.