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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. 1

%I #36 Jun 19 2022 23:23:22

%S 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 82,59,28,97,76,47,10,93,66,110,85,127,52,11,104,34,123,57,142,12,115,

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

%C It seems that all positive integers are included.

%C 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 Lindsey Witcosky, <a href="https://www.whitman.edu/Documents/Academics/Mathematics/SeniorProject_LindseyWitcosky.pdf">Perimeters of primitive Pythagorean triangles</a>

%H <a href="/index/Ps#PyTrip">Index entries related to Pythagorean Triples</a>.

%e From _Michel Marcus_, Mar 07 2018: (Start)

%e The first 10 terms of A081859 are 3, 5, 8, 7, 20, 12, 9, 28, 11, 16;

%e The first 10 terms of A081872 are 4, 12, 15, 24, 21, 35, 40, 45, 60, 63;

%e So the first 10 odd legs are 3, 5, 15, 7, 21, 35, 9, 45, 11, 63;

%e So the first 10 terms are 1, 2, 7, 3, 10, 17, 4, 22, 5, 31. (End)

%t (* lists a0* have to be prepared before *)

%t opPT = {a020882, a046087, a046086, a020882 + a046087 + a046086};

%t topPT = Transpose[opPT]; stopPT = SortBy[topPT, {#[[4]]} &];

%t tstopPT = Transpose[stopPT]; nopPT = tstopPT;

%t Do[ If[OddQ[tstopPT[[2]][[k]]], nopPT[[2]][[k]] = tstopPT[[2]][[k]];

%t nopPT[[3]][[k]] = tstopPT[[3]][[k]], nopPT[[2]][[k]] = tstopPT[[3]][[k]];

%t nopPT[[3]][[k]] = tstopPT[[2]][[k]]], {k, 1, 10000}];(nopPT[[2]] - 1)/2

%Y Cf. A297878 (even edge /4), A081872(b), A081859(c).

%Y Cf. A180620 (odd legs sorted on hypotenuse).

%K nonn

%O 1,2

%A _Ralf Steiner_, Jan 11 2018

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