A362290 a(n) is the number of parts into which the inner region of the parabola y = x^2 is divided by n squares inscribed in the parabola as described in the comments.

4, 8, 13, 19, 27, 35, 44, 54, 64, 76, 88, 100, 113, 127, 141, 155, 171, 187, 203, 219, 236, 254, 272, 290, 308, 328, 348, 368, 388, 408, 429, 451, 473, 495, 517, 539, 563, 587, 611, 635, 659, 683, 708, 734, 760, 786, 812, 838, 864, 892, 920, 948, 976, 1004, 1032, 1060, 1089, 1119, 1149

Offset

1,1

Comments

In the plane, consider the parabola y=x^2. There are n squares "strung" on the y axis so that two vertices of each square lie on the axis of the parabola, and the other two belong to the parabola. In this case, the sizes of the squares are chosen so that the lower vertices of the squares have ordinates 0, 1, 2, 3, ..., n - 1. See link.

Links

Table of n, a(n) for n=1..59.

Nicolay Avilov, Illustration of initial terms

Nicolay Avilov, Problem 2481. Squares in a parabola (in Russian).

Formula

a(n) = 2*(Sum_{k=1..n} floor(sqrt(4*k-3))) + 2 - floor((sqrt(4*n-3)-1)/2). - Oleg Sorokin, Apr 22 2023

Example

a(1) = 1 + 3 = 4;
a(2) = 2 + 6 = 8;
a(3) = 4 + 9 = 13.
See link.

Prog

(Python)
from math import isqrt
a, M = 3, []
for N in range(200):
   k = 4 * N + 1
   G = isqrt(k)
   a += 2 * G - (G ** 2 == k)
   M.append(a)
print(M) # Oleg Sorokin, Apr 22 2023

Crossrefs

Sequence in context: A312211 A034856 A365700 * A183865 A064609 A327566

Adjacent sequences: A362287 A362288 A362289 * A362291 A362292 A362293

Keyword

nonn

Author

Nicolay Avilov, Apr 14 2023

Status

approved