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".
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, 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, 2, 7, 1, 7, 1, 2, 4, 2, 3, 6, 1, 5, 2, 2, 1, 10, 2, 2, 2
Offset: 1
Keywords
Examples
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).
References
- 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.
Links
- Felix Huber, Table of n, a(n) for n = 1..10000
- Felix Huber, Cube with inscribed rectangle
- Felix Huber, Maple program to calculate the rectangles
Programs
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Maple
A383721:=proc(n) local a,x; a:=0; for x to 4*n-1 do if issqr(x^2+8*n^2) then a:=a+1 fi od; return a end proc; seq(A383721(n),n=1..87);
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Mathematica
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 *)
Formula
a(n) >= 1 since for x = n the rectangle is a square with integer area 18*n^2.
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