A249869 - OEIS

A249869

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

%I #27 Nov 28 2016 06:36:05

%S 6,0,30,60,0,84,0,210,0,180,210,0,0,0,330,0,630,0,924,0,546,504,0,

%T 1320,0,1560,0,840,0,1386,0,2340,0,0,0,1224,990,0,2730,0,0,0,3570,0,

%U 1710,0,2574,0,4620,0,5610,0,5016,0,2310,1716,0,0,0,7140,0,7980,0,0,0,3036

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

%C See A249866 for comments and references.

%C For the sorted areas of all primitive Pythagorean triangles (x, y, z) with, say y even, see A024406.

%C 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.

%C 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

%H Wolfdieter Lang, <a href="/A249869/a249869.pdf">First rows of the triangle.</a>

%H Wolfdieter Lang, <a href="/A249869/a249869_2.pdf">A Note on the Area table A249869 for Primitive Pythagorean Triangles.</a>

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

%e The triangle T(n, m) begins:

%e n\m 1 2 3 4 5 6 7 8 9 10 11

%e 2: 6

%e 3: 0 30

%e 4: 60 0 84

%e 5: 0 210 0 180

%e 6: 210 0 0 0 330

%e 7: 0 630 0 924 0 546

%e 8: 504 0 1320 0 1560 0 840

%e 9: 0 1386 0 2340 0 0 0 1224

%e 10: 990 0 2730 0 0 0 3570 0 1710

%e 11: 0 2574 0 4620 0 5610 0 5016 0 2310

%e 12: 1716 0 0 0 7140 0 7980 0 0 0 3036

%e ...

%e For more rows see the link.

%e T(5, 2) = 210 for the primitive triangle (21, 20, 29).

%e T(6, 1) = 210 for the primitive triangle (35, 12, 37).

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

%K nonn,easy,tabl

%O 2,1

%A _Wolfdieter Lang_, Dec 03 2014