A286637 Perimeter of primitive integer automedian triangles (without repetitions).

3, 37, 47, 73, 107, 121, 181, 191, 239, 249, 253, 299, 337, 359, 429, 431, 433, 503, 529, 537, 541, 579, 587, 649, 659, 661, 671, 759, 767, 781, 789, 793, 851, 863, 933, 937, 971, 1067, 1069, 1079, 1081, 1093, 1191, 1199, 1237, 1249, 1257, 1261, 1307, 1319

Offset

1,1

Comments

An automedian triangle has the square of its sides in an arithmetic progression. Hence its sides (a, b, c) with a <= b <= c are related by a^2+c^2 = 2b^2. Also a+b > c in order to satisfy the triangle inequality. It is so named because its sides and medians are proportional to each other but in a different order. Apart from the unit equilateral triangle all other primitive integer automedian triangles can be generated from primitive Pythagorean triangles (PPT's). If x^2+y^2 = z^2 is a PPT then a = |x-y|, b = z and c = x+y gives an integer automedian triangle (a, b, c) provided that Min(2x, 2y) < z. Note that the PPT (3, 4, 5) will not generate an integer automedian triangle.

Links

Table of n, a(n) for n=1..50.

Ralph H. Buchholz, On Triangles with rational altitudes, angle bisectors or medians (p35)

C. F. Parry, Steiner-Lehmus and the Automedian Triangle

Wikipedia, Automedian triangle.

Formula

Using the Euclidean parameters m, n to generate a PPT where (x, y, z) = (m^2-n^2, 2m*n, m^2+n^2) with m > n, GCD(m, n) = 1 and m+n odd, we can also generate an integer automedian triangle (a, b, c) = (|m^2-n^2-2m*n|, m^2+n^2, m^2-n^2+2m*n) with additionally n < m < n*Sqrt(3) or m > n*(2+Sqrt(3)) to satisfy the triangle inequality.

Example

Substituting m=3 and n=2 into the Euclidean parameters in the above formula it gives PPT (5, 12, 13) and it will generate an integer automedian triangle (7, 13, 17) with perimeter 37. It is the second occurrence of a primitive integer automedian triangle.

Mathematica

lst={}; Do[{a, b, c}={Abs[m^2-n^2-2m*n], m^2+n^2, m^2-n^2+2m*n}; If[GCD[m, n]==1&&OddQ[m+n]&&a+b>c, AppendTo[lst, a+b+c]], {m, 1, 100}, {n, 0, m}]; Take[Union@lst, 50]

Crossrefs

Sequence in context: A042333 A106103 A224471 * A106996 A338751 A300347

Adjacent sequences: A286634 A286635 A286636 * A286638 A286639 A286640

Keyword

nonn

Author

Frank M Jackson, May 11 2017

Status

approved