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A374597 a(n) = floor(area) for the area of the largest square that can be inscribed in the n-th Pythagorean triangle, with one side of the square on the hypotenuse of the triangle, for Pythagorean triangles ordered first by increasing perimeter, then by shorter leg.

%I A374597 #65 Apr 10 2025 08:04:18
%S A374597 2,10,11,23,24,42,28,46,65,93,94,99,75,128,52,104,168,213,112,185,223,
%T A374597 262,269,84,318,373,156,378,290,391,444,398,252,301,515,584,209,417,
%U A374597 591,124,673,555,621,759,632,568,839,852,269,448,949,1038,172,742,895,1051,679,1077
%N A374597 a(n) = floor(area) for the area of the largest square that can be inscribed in the n-th Pythagorean triangle, with one side of the square on the hypotenuse of the triangle, for Pythagorean triangles ordered first by increasing perimeter, then by shorter leg.
%C A374597 For a triangle with leg lengths x,y, the square has side length x*y*z/(x*y + z^2) and the area rounded down is a(n) = f(x,y,z) = floor((x*y*z/(x*y + z^2))^2) .
%H A374597 Alexander M. Domashenko, <a href="https://www.diofant.ru/problem/3957/">Problem 2176. Two squares in a triangle</a> (in Russian).
%e A374597 The first Pythagorean triangle is (x,y,z) = (3,4,5) and the rounded area of the square inside it is a(1) = f(3,4,5) = floor((3*4*5/(3*4+5^2))^2) = 2.
%Y A374597 Cf. A376608.
%K A374597 nonn
%O A374597 1,1
%A A374597 _Alexander M. Domashenko_, Jul 13 2024