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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.

%I A088544 #19 Nov 28 2023 22:05:28
%S A088544 37,229,409,793,1261,2041,1789,4381,5233,4069,8317,6073,14449,7969,
%T A088544 12181,9997,11041,23473,14089,24457,17341,36181,20773,53461,29341,
%U A088544 44269,28009,38509,76297,35869,44257,74209,42841,105769,50137,65701,53209
%N 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.
%C A088544 Such an inscribed square has side x*y*z = A063011(n).
%C A088544 Also the radius squared of the Conway circle of a primitive Pythagorean triangle, sorted on product of sides. - _Frank M Jackson_, Nov 04 2023
%D A088544 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.
%H A088544 Frank M Jackson, <a href="/A088544/b088544.txt">Table of n, a(n) for n = 1..10000</a>
%H A088544 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/ConwayCircle.html">MathWorld: Conway circle</a>.
%H A088544 Wikipedia, <a href="https://en.wikipedia.org/wiki/Conway_circle_theorem">Conway circle theorem</a>.
%F A088544 a(n) = x*y + z^2.
%F A088544 a(n) = s^2 + r^2, where s is the semiperimeter and r is the inradius of triangle (x, y, z).
%t A088544 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 *)
%Y A088544 Cf. A088509, A088510, A088511, A063011.
%K A088544 nonn
%O A088544 1,1
%A A088544 _Lekraj Beedassy_, Nov 17 2003
%E A088544 More terms from _Max Alekseyev_, May 30 2009