A365316 a(n) = number of visible diamonds (squares rotated by 45 degrees) with diagonal 2 in the large square bounded by the parabola y=x^2 and starting from the point (0, 2n).

1, 5, 8, 16, 21, 26, 38, 45, 52, 59, 75, 84, 93, 102, 111, 131, 142, 153, 164, 175, 186, 210, 223, 236, 249, 262, 275, 288, 316, 331, 346, 361, 376, 391, 406, 421, 453, 470, 487, 504, 521, 538, 555, 572, 589, 625, 644, 663, 682, 701, 720, 739, 758, 777, 796, 836

Offset

0,2

Links

Table of n, a(n) for n=0..55.

Anatoliy A. Abramov, Squares and parabola visualization.

Formula

a(n) = f(n) + a(n - 1), where f(n) = Sum_{i=size(n-1)..size(n)} 2*i - 1 and size(n) = floor((1 + sqrt(1 + 8*n) / 2).

Example

Illustration for n = 0..3:
.        _____________________________
       /\                           ^
      /  \                          |
     /\ d/\                         |
    /  \/  \                        |
   /\ d/\ d/\              vertical |
  /  \/  \/  \              limits  |
  \ d/\ d/\ d/             of large |
   \/  \/  \/____________  square D |
    \ d/\ d/           ^            |
     \/  \/   vertical |            |
     /\ c/\    limits  |            |
    /  \/  \ _ _ of _ _|____________v_
    \ c/\ c/   large   |            ^
     \/  \/   square C |            |
     /\ b/\            |   vertical |
    /  \/  \ __________v_   limits  |
    \ b/\ b/               of large |
     \/  \/                square B |
      \ b/                          |
       \/___________________________v_
       /\      limits  ^
      /  \    of large |
      \ a/    square A |
       \/______________v_
.
n=0: Large square A builds from its bottom vertex at (0, 0), has diagonal size 1*2 = 2 with lateral vertices at (-1, 1) and (1, 1), and is bounded by the parabola at (-1, 1) and (1, 1). It coincides with the 1 small square a. We have 1 small square thus far, so a(0) = 1.
n=1: Large square B builds from its bottom vertex at (0, 2), has diagonal size 2*2 = 4 with lateral vertices at (-2, 4) and (2, 4), and is bounded by the parabola at (-2, 2) and (2, 2). It contains the 4 small squares b. The total number of small squares thus far is a(1) = 1 + 4 = 5.
n=2: Large square C builds from its bottom vertex at (0, 4), has diagonal size 2*2 = 4 with lateral vertices at (-2, 6) and (2, 6), and is bounded by the parabola at (-2, sqrt(6)) and (2, sqrt(6)). It contains the 3 small squares c as well as the topmost small square b (which has already been counted). The total number of small squares thus far is a(2) = 1 + 4 + 3 = 8.
n=3: Large square D builds from its bottom vertex at (0, 6), has diagonal size 3*2 = 6 with lateral vertices at (-3, 9) and (3, 9), and is bounded by the parabola at (-3, 9) and (3, 9). It contains the 8 small squares d as well as the topmost small square c (which has already been counted). The total number of small squares thus far is a(3) = 1 + 4 + 3 + 8 = 16.

Prog

(Java)
import java.util.stream.IntStream;
public class Main {
    public static void main(String[] args) {
        IntStream.rangeClosed(0, 100).forEach(it -> System.out.printf("%d, ", sumAllVisibleSquare(it)));
    }
    private static int maxSquareSize(int n) {
        return (int) Math.floor((1 + Math.sqrt(1 + 8 * n)) / 2);
    }
    private static int sumVisibleSquares(int n) {
        int upSquareSize = maxSquareSize(n);
        int lowSquareSize = maxSquareSize(n - 1);
        return IntStream.rangeClosed(lowSquareSize, upSquareSize).map(it -> it + (it - 1)).sum();
    }
    private static int sumAllVisibleSquare(int n) {
        return n == 0 ? 1 : sumVisibleSquares(n) + sumAllVisibleSquare(n - 1);
    }
}

Crossrefs

Sequence in context: A314564 A314565 A126695 * A001082 A242090 A030006

Adjacent sequences: A365313 A365314 A365315 * A365317 A365318 A365319

Keyword

nonn

Author

Anatoliy A. Abramov, Sep 01 2023

Status

approved