A362869 a(n) is equal to the number of cells in one octant of the octagon of unit squares with side equal n.

1, 2, 8, 11, 22, 27, 43, 50, 71, 80, 106, 117, 148, 161, 197, 212, 253, 270, 316, 335, 386, 407, 463, 486, 547, 572, 638, 665, 736, 765, 841, 872, 953, 986, 1072, 1107, 1198, 1235, 1331, 1370, 1471, 1512, 1618, 1661, 1772, 1817, 1933, 1980, 2101, 2150, 2276

Offset

1,2

Comments

The octagon is obtained from a square of side 3n-2 cells by removing a triangle of cells from each of the four corners.

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Anatoliy A. Abramov, Illustration of initial terms

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Formula

a(n) = k*(k + 1)/2 - floor(n^2/4), where k = ceiling((n + 2*(n - 1))/2).

G.f.: x*(1 + x + 4*x^2 + x^3)/((1 - x)^3*(1 + x)^2). - Stefano Spezia, May 07 2023

Example

For n=3, the octagon is a square of side 3n-2 = 7 with corners chopped to leave each side with 3 "*",
        * * *
      * * * * *
    * * * * * * *
    @ @ @ @ * * *
    @ @ @ * * * *
      @ * * * *
        * * *
The a(3) = 8 cells marked "@" are an eighth of the figure and are the primitive cells in the sense that other cells map to them by rotations or reflections.

Mathematica

LinearRecurrence[{1, 2, -2, -1, 1}, {1, 2, 8, 11, 22}, 100] (* Paolo Xausa, Oct 16 2023 *)

Prog

(Java)
import java.util.function.Function;
import java.util.stream.IntStream;
public class Main {
    private static int A362869(int n) {
        Function<Integer, Integer> trianglarNumber = p -> p * (p + 1) / 2;
        Function<Integer, Integer> quarterOfSquareNumber = p -> p * p / 4;
        int k = (n + 2 * (n - 1) + 1) / 2;
        return trianglarNumber.apply(k) - quarterOfSquareNumber.apply(n);
    }
    public static void main(String[] args) {
        IntStream.range(1, 100).forEach(n -> System.out.print(A362869(n) + ", "));
    }
}

Crossrefs

Cf. A000217, A002620.

Sequence in context: A009420 A189328 A090746 * A234924 A336771 A174114

Adjacent sequences: A362866 A362867 A362868 * A362870 A362871 A362872

Keyword

nonn,easy

Author

Anatoliy A. Abramov, May 07 2023

Status

approved