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A297878 1/4 of the even edges of primitive Pythagorean triangles with legs (b=A081872, c=A081859), ordered by semiperimeters.

%I A297878 #54 Jun 19 2022 23:23:18
%S A297878 1,3,2,6,5,3,10,7,15,4,14,12,21,9,20,5,18,28,15,11,36,6,22,35,33,45,
%T A297878 13,30,44,7,26,42,55,21,39,15,35,8,52,66,30,65,24,63,17,78,40,9,60,77,
%U A297878 34,56,91,51,19,72,45,10,68,88,105,38,63,85,30,104,57,21,102,120,11,76,99,42,119,70,33,95
%N A297878 1/4 of the even edges of primitive Pythagorean triangles with legs (b=A081872, c=A081859), ordered by semiperimeters.
%C A297878 It seems that all positive integers are included.
%C A297878 Every term has the form of edge length e = (v-u)*u/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 A297878 Wikipedia, <a href="https://en.wikipedia.org/wiki/Pythagorean_triple">Pythagorean triple</a>.
%H A297878 Lindsey Witcosky, <a href="https://www.whitman.edu/Documents/Academics/Mathematics/SeniorProject_LindseyWitcosky.pdf">Perimeters of primitive Pythagorean triangles</a>.
%H A297878 <a href="/index/Ps#PyTrip">Index entries related to Pythagorean Triples</a>.
%e A297878 From _Michel Marcus_, Mar 07 2018: (Start)
%e A297878 The first 10 terms of A081859 are 3,  5,  8,  7, 20, 12,  9, 28, 11, 16;
%e A297878 The first 10 terms of A081872 are 4, 12, 15, 24, 21, 35, 40, 45, 60, 63;
%e A297878 So the first 10 even legs are     4, 12,  8, 24, 20, 12, 40, 28, 60, 16;
%e A297878 So the first 10 terms are         1,  3,  2,  6,  5,  3, 10,  7, 15,  4. (End)
%t A297878 (* lists a0* have to be prepared before *)
%t A297878 opPT = {a020882, a046087, a046086, a020882 + a046087 + a046086} topPT = Transpose[opPT]; stopPT = SortBy[topPT, {#[[4]]} &]; tstopPT = Transpose[stopPT]; nopPT = tstopPT; Do[ If[OddQ[tstopPT[[2]][[k]]], nopPT[[2]][[k]] = tstopPT[[2]][[k]]; nopPT[[3]][[k]] = tstopPT[[3]][[k]], nopPT[[2]][[k]] = tstopPT[[3]][[k]]; nopPT[[3]][[k]] = tstopPT[[2]][[k]]], {k, 1, 10000}]; nopPT[[3]]/4
%Y A297878 Cf. A298042((odd edge - 1)/2), A081872(b), A081859(c).
%Y A297878 Cf. A231100 (even legs ordered by hypotenuse).
%K A297878 nonn
%O A297878 1,2
%A A297878 _Ralf Steiner_, Jan 07 2018