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

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, 13, 30, 44, 7, 26, 42, 55, 21, 39, 15, 35, 8, 52, 66, 30, 65, 24, 63, 17, 78, 40, 9, 60, 77, 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

Offset: 1

Author

Ralf Steiner, Jan 07 2018

Keywords

• nonn

Comments

It seems that all positive integers are included.

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).

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 even legs are     4, 12,  8, 24, 20, 12, 40, 28, 60, 16;
So the first 10 terms are         1,  3,  2,  6,  5,  3, 10,  7, 15,  4. (End)
		

Links

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

Crossrefs

Cf. A298042((odd edge - 1)/2), A081872(b), A081859(c).

Cf. A231100 (even legs ordered by 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[[3]]/4