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

Author

Gonzalo Martínez, Aug 17 2024

Keywords

• nonn

Comments

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)).

Crossrefs

Cf. A361795, A000194.

Formula

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