A088544 Scale factor by which primitive Pythagorean triangle {x=A088509(n), y=A088510(n), z=A088511(n)} needs be enlarged in order to circumscribe the smallest integral square having a side on the hypotenuse.

37, 229, 409, 793, 1261, 2041, 1789, 4381, 5233, 4069, 8317, 6073, 14449, 7969, 12181, 9997, 11041, 23473, 14089, 24457, 17341, 36181, 20773, 53461, 29341, 44269, 28009, 38509, 76297, 35869, 44257, 74209, 42841, 105769, 50137, 65701, 53209

Offset: 1

Author

Lekraj Beedassy, Nov 17 2003

Keywords

• nonn

Comments

Such an inscribed square has side x*y*z = A063011(n).

Also the radius squared of the Conway circle of a primitive Pythagorean triangle, sorted on product of sides. - Frank M Jackson, Nov 04 2023

References

• J. D. E. Konhauser et al., Which Way Did The Bicycle Go?, Problem 21, "The Square on the Hypotenuse", pp. 7; 79-80, Dolciani Math. Exp. No. 18, MAA, 1996.

Links

• Frank M Jackson, Table of n, a(n) for n = 1..10000
• Eric W. Weisstein, MathWorld: Conway circle.
• Wikipedia, Conway circle theorem.

Crossrefs

Cf. A088509, A088510, A088511, A063011.

Programs

Mathematica

lst={}; k=25; Do[If[GCD[m, n]==1&&OddQ[m+n], AppendTo[lst, {2m*n(m^4-n^4), m^2(m+n)^2+n^2(m-n)^2}]], {m, 1, k}, {n, 1, m}]; lst=Sort@lst; Table[lst[[n]][[2]], {n, 1, 100}] (* Frank M Jackson, Nov 04 2023 *)

Formula

a(n) = x*y + z^2.

a(n) = s^2 + r^2, where s is the semiperimeter and r is the inradius of triangle (x, y, z).

Extensions

More terms from Max Alekseyev, May 30 2009