A361795 -id:A361795 - cp's OEIS frontend

A375473 a(n) is the area of the largest rectangle with integer sides that can be inscribed under the parabola y = -x^2 + n and on or above the x-axis.

0, 0, 2, 4, 6, 8, 10, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336

Offset: 0

Gonzalo Martínez, Aug 17 2024

nonn

Given the function defined by f(x) = -x^2 + n, the area of each rectangle inscribed under the parabola associated with f and on the x-axis is modeled by the function g(x) = 2x*(-x^2 + n), where 2x is the base of the rectangle and ( -x^2 +n) is its height. The value of x that maximizes the area is x = sqrt(n/3). However, this value is not always an integer, so x should be chosen as the nearest integer to sqrt(n/3), which corresponds to floor(1/2 + sqrt(n/3 - 1/12)).

Cf. A361795, A000194.

a(n) = 2*floor(1/2 + sqrt(n/3 - 1/12))*(-(floor(1/2 + sqrt(n/3 - 1/12)))^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".

Original entry on oeis.org

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

Felix Huber, May 08 2025

nonn

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).
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.
Conjecture: There is no rectangle with an integer area if x is not a positive integer.

			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).

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.

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

Cf. A361795, A373710, A375473.

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);

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 *)

a(n) >= 1 since for x = n the rectangle is a square with integer area 18*n^2.

cp's OEIS Frontend

A375473 a(n) is the area of the largest rectangle with integer sides that can be inscribed under the parabola y = -x^2 + n and on or above the x-axis.

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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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