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A383721 a(n) is the number of distinct rectangles with integer area that can be inscribed in a cube with edge length 4*n, as shown in the linked figure "Cube with inscribed rectangle".

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%I A383721 #10 May 14 2025 00:00:31
%S A383721 1,2,2,2,1,4,1,2,2,3,1,5,1,3,4,2,1,4,1,4,3,2,1,5,1,2,2,4,1,8,1,2,3,2,
%T A383721 3,6,1,2,3,5,1,7,1,3,6,2,1,5,1,3,2,3,1,4,3,5,2,2,1,11,1,2,5,2,2,6,1,3,
%U A383721 2,7,1,7,1,2,4,2,3,6,1,5,2,2,1,10,2,2,2
%N A383721 a(n) is the number of distinct rectangles with integer area that can be inscribed in a cube with edge length 4*n, as shown in the linked figure "Cube with inscribed rectangle".
%C A383721 See linked figure "Cube with inscribed rectangle": The inscribed quadrilateral is a rectangle for all choices of x because it is point-symmetric with respect to the center of the cube and its two diagonals are of equal length. Its side lengths are sqrt(2)*(4*n - x) and sqrt(2*x^2 + 16*n^2), its area is 2*(4*n - x)*sqrt(x^2 + 8*n^2).
%C A383721 a(n) is also the number of solutions to x^2 + 8*n^2 = y^2 in positive integers x and y where 1 <= x <= 4*n - 1.
%C A383721 Conjecture: There is no rectangle with an integer area if x is not a positive integer.
%D A383721 Michael Graf and Heinz Klemenz, Geometry 2 Exercises (second edition), Swiss-German Mathematics Commission, hep Verlag, Bern, 2021, chapter 9.3, exercise 24, p. 71.
%H A383721 Felix Huber, <a href="/A383721/b383721.txt">Table of n, a(n) for n = 1..10000</a>
%H A383721 Felix Huber, <a href="/A383721/a383721.pdf">Cube with inscribed rectangle</a>
%H A383721 Felix Huber, <a href="/A383721/a383721.txt">Maple program to calculate the rectangles</a>
%F A383721 a(n) >= 1 since for x = n the rectangle is a square with integer area 18*n^2.
%e A383721 The a(6) = 4 rectangles with an integer area, which can be inscribed in a cube with the edge length 4*6 = 24, arise for x = 1 (side lengths 23*sqrt(2) and 17*sqrt(2), area 782), x = 6 (square with side length 18*sqrt(2), area 648), x = 14 (10*sqrt(2) and 22*sqrt(2), area 440) and x = 21 (side lengths 3*sqrt(2) and 27*sqrt(2), area 162).
%p A383721 A383721:=proc(n)
%p A383721     local a,x;
%p A383721     a:=0;
%p A383721     for x to 4*n-1 do
%p A383721         if issqr(x^2+8*n^2) then
%p A383721             a:=a+1
%p A383721         fi
%p A383721     od;
%p A383721     return a
%p A383721 end proc;
%p A383721 seq(A383721(n),n=1..87);
%t A383721 A383721[n_] := Module[{i = 0, x}, For[x = 1, x <= 4n - 1, x++, If[IntegerQ[Sqrt[x^2 + 8n^2]], i++] ]; i ];Array[A383721,87] (* _James C. McMahon_, May 13 2025 *)
%Y A383721 Cf. A361795, A373710, A375473.
%K A383721 nonn
%O A383721 1,2
%A A383721 _Felix Huber_, May 08 2025