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

%I A249869 #27 Nov 28 2016 06:36:05
%S A249869 6,0,30,60,0,84,0,210,0,180,210,0,0,0,330,0,630,0,924,0,546,504,0,
%T A249869 1320,0,1560,0,840,0,1386,0,2340,0,0,0,1224,990,0,2730,0,0,0,3570,0,
%U A249869 1710,0,2574,0,4620,0,5610,0,5016,0,2310,1716,0,0,0,7140,0,7980,0,0,0,3036
%N A249869 Triangle giving the area of primitive Pythagorean triangles, with zero entries for non-primitive triangles.
%C A249869 See A249866 for comments and references.
%C A249869 For the sorted areas of all primitive Pythagorean triangles (x, y, z) with, say y even, see A024406.
%C A249869 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 A249869 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 A249869 Wolfdieter Lang, <a href="/A249869/a249869.pdf">First rows of the triangle.</a>
%H A249869 Wolfdieter Lang, <a href="/A249869/a249869_2.pdf">A Note on the Area table A249869 for Primitive Pythagorean Triangles.</a>
%F A249869 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 A249869 The triangle T(n, m) begins:
%e A249869 n\m    1    2    3     4     5     6    7     8     9   10    11
%e A249869 2:     6
%e A249869 3:     0   30
%e A249869 4:    60    0   84
%e A249869 5:     0  210    0   180
%e A249869 6:   210    0    0     0   330
%e A249869 7:     0  630    0   924     0   546
%e A249869 8:   504    0 1320     0  1560     0  840
%e A249869 9:     0 1386    0  2340     0     0    0 1224
%e A249869 10:  990    0 2730     0     0     0 3570    0   1710
%e A249869 11:    0 2574    0  4620     0  5610    0 5016      0 2310
%e A249869 12: 1716    0    0     0  7140     0 7980     0     0    0  3036
%e A249869 ...
%e A249869 For more rows see the link.
%e A249869 T(5, 2) = 210 for the primitive triangle (21, 20, 29).
%e A249869 T(6, 1) = 210 for the primitive triangle (35, 12, 37).
%Y A249869 Cf. A024406, A249866, A258150 (one sixth of this triangle), A225949 (leg sums), A225951 (perimeters), A222946 (hypotenuses), A208854 (odd catheti), A208855 (even catheti), A278711.
%K A249869 nonn,easy,tabl
%O A249869 2,1
%A A249869 _Wolfdieter Lang_, Dec 03 2014