A278711 -id:A278711 - cp's OEIS frontend

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

Wolfdieter Lang, Dec 03 2014

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

			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).

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

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

T(n, m) = n*m*(n+m)(n-m) if n > m >= 1, (-1)^(n+m) = -1 and gcd(n,m) = 1, else 0.

A278712 Triangle T read by rows: T(n, m), for n >= 2, and m = 1, 2, ..., n-1, equals the square root of the positive integer solution y of y^2 = x^3 - A(n, m)^2*x with the area A(n, m) = A249869(n, m) of the primitive Pythagorean triangle characterized by (n, m) or 0 if no such triangle exists.

Original entry on oeis.org

6, 0, 15, 60, 0, 28, 0, 105, 0, 45, 210, 0, 0, 0, 66, 0, 315, 0, 231, 0, 91, 504, 0, 440, 0, 312, 0, 120, 0, 693, 0, 585, 0, 0, 0, 153, 990, 0, 910, 0, 0, 0, 510, 0, 190, 0, 1287, 0, 1155, 0, 935, 0, 627, 0, 231, 1716, 0, 0, 0, 1428, 0, 1140, 0, 0, 0, 276, 0, 2145, 0, 1989, 0, 1729, 0, 1365, 0, 897, 0, 325, 2730, 0, 2618, 0, 2394, 0, 0, 0, 1610, 0, 1050, 0, 378, 0, 3315, 0, 3135, 0, 0, 0, 2415, 0, 0, 0, 0, 0, 435

Offset: 2

Wolfdieter Lang, Nov 27 2016

The corresponding solutions x are given in A278711, where also details are found.

			The triangle T(n, m) begins:
n\m   1    2   3    4   5   6   7   8   9  10
2:    6
3:    0   15
4:   60    0  28
5:    0  105   0   45
6:  210    0   0    0  66
7:    0  315   0  231   0  91
8:  504    0 440    0 312   0 120
9:    0  693   0  585   0   0   0 153
10: 990    0 910    0   0   0 510   0 190
11:   0 1287   0 1155   0 935   0 627   0 231
...
n = 12: 1716 0 0 0 1428 0 1140 0 0 0 276,
n = 13: 0 2145 0 1989 0 1729 0 1365 0 897 0 325,
n = 14: 2730 0 2618 0 2394 0 0 0 1610 0 1050 0 378,
n = 15: 0 3315 0 3135 0 0 0 2415 0 0 0 0 0 435.
...
For the solutions [x,y] see A278711.

Cf. A278711.

T(n, m) = (n^2 - m^2)*n if n > m >= 1, gcd(n, m) = 1 and n+m is odd, and T(n, m) = 0 otherwise.

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

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