This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%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