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A298042 (d-1)/2 of the odd edges d of primitive Pythagorean triangles with legs (b=A081872, c=A081859), ordered by semiperimeters.

%I A298042 #36 Jun 19 2022 23:23:22
%S A298042 1,2,7,3,10,17,4,22,5,31,16,27,6,38,19,49,32,7,45,58,8,71,52,25,42,9,
%T A298042 82,59,28,97,76,47,10,93,66,110,85,127,52,11,104,34,123,57,142,12,115,
%U A298042 161,80,37,136,103,13,126,178,87,149,199,112,67,14,172,137,94,195,43,162,218,72,15,241
%N A298042 (d-1)/2 of the odd edges d of primitive Pythagorean triangles with legs (b=A081872, c=A081859), ordered by semiperimeters.
%C A298042 It seems that all positive integers are included.
%C A298042 Every term is equal to (d-1)/2 with d = 2*u*v - v^2, semiperimeter s = (h+b+c)/2 = u*v with b > c, h^2 = b^2 + c^2, u < v < 2*u, v odd (see Theorem 3 of Witcosky).
%H A298042 Lindsey Witcosky, <a href="https://www.whitman.edu/Documents/Academics/Mathematics/SeniorProject_LindseyWitcosky.pdf">Perimeters of primitive Pythagorean triangles</a>
%H A298042 <a href="/index/Ps#PyTrip">Index entries related to Pythagorean Triples</a>.
%e A298042 From _Michel Marcus_, Mar 07 2018: (Start)
%e A298042 The first 10 terms of A081859 are 3,  5,  8,  7, 20, 12,  9, 28, 11, 16;
%e A298042 The first 10 terms of A081872 are 4, 12, 15, 24, 21, 35, 40, 45, 60, 63;
%e A298042 So the first 10 odd legs are      3,  5, 15,  7, 21, 35,  9, 45, 11, 63;
%e A298042 So the first 10 terms are         1,  2,  7,  3, 10, 17,  4, 22,  5, 31. (End)
%t A298042 (* lists a0* have to be prepared before *)
%t A298042 opPT = {a020882, a046087, a046086, a020882 + a046087 + a046086};
%t A298042 topPT = Transpose[opPT]; stopPT = SortBy[topPT, {#[[4]]} &];
%t A298042 tstopPT = Transpose[stopPT]; nopPT = tstopPT;
%t A298042 Do[ If[OddQ[tstopPT[[2]][[k]]], nopPT[[2]][[k]] = tstopPT[[2]][[k]];
%t A298042 nopPT[[3]][[k]] = tstopPT[[3]][[k]], nopPT[[2]][[k]] = tstopPT[[3]][[k]];
%t A298042 nopPT[[3]][[k]] = tstopPT[[2]][[k]]], {k, 1, 10000}];(nopPT[[2]] - 1)/2
%Y A298042 Cf. A297878 (even edge /4), A081872(b), A081859(c).
%Y A298042 Cf. A180620 (odd legs sorted on hypotenuse).
%K A298042 nonn
%O A298042 1,2
%A A298042 _Ralf Steiner_, Jan 11 2018