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

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, 82, 59, 28, 97, 76, 47, 10, 93, 66, 110, 85, 127, 52, 11, 104, 34, 123, 57, 142, 12, 115, 161, 80, 37, 136, 103, 13, 126, 178, 87, 149, 199, 112, 67, 14, 172, 137, 94, 195, 43, 162, 218, 72, 15, 241

Offset: 1

Author

Ralf Steiner, Jan 11 2018

Keywords

• nonn

Comments

It seems that all positive integers are included.

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).

Examples

			From _Michel Marcus_, Mar 07 2018: (Start)
The first 10 terms of A081859 are 3,  5,  8,  7, 20, 12,  9, 28, 11, 16;
The first 10 terms of A081872 are 4, 12, 15, 24, 21, 35, 40, 45, 60, 63;
So the first 10 odd legs are      3,  5, 15,  7, 21, 35,  9, 45, 11, 63;
So the first 10 terms are         1,  2,  7,  3, 10, 17,  4, 22,  5, 31. (End)
		

Links

• Lindsey Witcosky, Perimeters of primitive Pythagorean triangles
• Index entries related to Pythagorean Triples.

Crossrefs

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

Cf. A180620 (odd legs sorted on hypotenuse).

Programs

Mathematica

(* lists a0* have to be prepared before *)
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[[2]] - 1)/2