OFFSET
0,3
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).
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).
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
EXAMPLE
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.
MATHEMATICA
Floor[8/27*Range[0, 50]^3] (* Paolo Xausa, Jan 30 2025 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gonzalo Martínez and Javier Astudillo, Jan 04 2025
STATUS
approved