A379852 a(n) = floor(8*n^3/27).

0, 0, 2, 8, 18, 37, 64, 101, 151, 216, 296, 394, 512, 650, 813, 1000, 1213, 1455, 1728, 2032, 2370, 2744, 3154, 3605, 4096, 4629, 5207, 5832, 6504, 7226, 8000, 8826, 9709, 10648, 11645, 12703, 13824, 15008, 16258, 17576, 18962, 20421, 21952, 23557, 25239

Offset: 0

Author

Gonzalo Martínez and Javier Astudillo, Jan 04 2025

Keywords

• nonn
• easy

Comments

a(n) is the integer part of the area of the largest triangle that can be inscribed in the region bounded by the parabola y = x^2, the x-axis, and the line x = n.

To estimate the integral int_{x = 0..n} x^2 dx by means of a triangle, we find that the triangle with the largest area that can be inscribed in the region bounded by the parabola y = x^2, the x-axis and the line x = n is the right triangle with vertices (n/3, 0), (n, 0) and (n, (8/9)*n^2), whose area is (2n/3)^3 and a(n) has been defined as floor((2n/3)^3).

Examples

			If n = 2, the largest triangle that can be inscribed in the region bounded by the parabola y = x^2, the x-axis, and the line x = n is the right triangle with vertices (2/3,0),(2,0) and (2,32/9), whose area is 64/27. Since floor(64/27) = 2, it follows that a(2) = 2.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,0,0,1,-3,3,-1).

Crossrefs

Cf. A016743, A375473.

Programs

Mathematica

Floor[8/27*Range[0, 50]^3] (* Paolo Xausa, Jan 30 2025 *)

Formula

a(n) = floor(A016743(n)/27).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-9) - 3*a(n-10) + 3*a(n-11) - a(n-12) for n >= 12. - Pontus von Brömssen, Jan 14 2025