A small Java program for computing volumes of n-dimensional balls.

NdimensionalBallVolumes.java

package io.github.coderodde.math.simulation.volumes;

public class NdimensionalBallVolumes {

    private static final int LARGEST_N = 10;

    public static void main(String[] args) {
        System.out.println("$$");
        System.out.println("\\begin{aligned}");

        int maxLineNumberLength = Integer.toString(LARGEST_N).length();

        String lineFormat = 
                String.format(
                    "V_{%%%dd} &= %%-30s & \\\\ %%%% n = %%%dd\n",
                        maxLineNumberLength,
                        maxLineNumberLength);

        for (int n = 1; n <= LARGEST_N; ++n) {
            Volume v = getVolume(n);
            System.out.printf(lineFormat,
                              n,
                              v.toString(),
                              n);
        }

        System.out.println("\\end{aligned}");
        System.out.println("$$");
    }

    static Volume getVolume(int n) {
        if (n == 1) {
            return new Volume(2, 1, 0, n);
        }

        if (n == 2) {
            return new Volume(1, 1, 1, n);
        }

        Volume previous = getVolume(n - 2);

        return new Volume(previous.numerator   * 2,
                          previous.denominator * n,
                          previous.piExponent + 1,
                          n);
    }

    static final class Volume {

        long numerator;
        long denominator;
        int piExponent;
        int n;

        Volume(long numerator,
               long denominator,
               int piExponent,
               int n) {

            long gcd = gcd(numerator, 
                           denominator);

            this.numerator   = numerator   / gcd;
            this.denominator = denominator / gcd;
            this.piExponent  = piExponent;
            this.n           = n;
        }

        @Override
        public String toString() {
            StringBuilder sb = new StringBuilder();

            if (denominator == 1) {
                appendNumeratorString(sb);
            } else {
                sb.append("\\frac{");
                appendNumeratorString(sb);
                sb.append("}{");
                sb.append(denominator);
                sb.append("}");
            }

            return sb.toString();
        }

        private void appendNumeratorString(StringBuilder sb) {
            if (numerator > 1) {
                sb.append(numerator)
                  .append(" ");
            }

            if (piExponent > 0) {
                sb.append("\\pi");

                if (piExponent > 1) {
                    sb.append("^{")
                      .append(piExponent)
                      .append("}");
                }
            }

            sb.append(n == 1 ? "R" : " R");

            if (n > 1) {
                sb.append("^{")
                  .append(n)
                  .append("}");
            }
        }
    }

    static long gcd(long a, long b) {
        while (b != 0) {
            long t = b;
            b = a % b;
            a = t;
        }   

        return a;
    }
}

NdimensionalBallVolumesTest.java

  package io.github.coderodde.math.simulation.volumes;

import io.github.coderodde.math.simulation.volumes.NdimensionalBallVolumes.Volume;
import org.junit.Test;
import static org.junit.Assert.*;

public class NdimensionalBallVolumesTest {
    
    @Test
    public void n1() {
        Volume f = NdimensionalBallVolumes.getVolume(1);
        
        assertEquals(2, f.numerator);
        assertEquals(1, f.denominator);
        assertEquals(0, f.pis);
        String s = f.toString();
        System.out.println(s);
    }
    
    @Test
    public void n2() {
        Volume f = NdimensionalBallVolumes.getVolume(2);
        
        assertEquals(1, f.numerator);
        assertEquals(1, f.denominator);
        assertEquals(1, f.pis);
        String s = f.toString();
        System.out.println(s);
    }
    
    @Test
    public void n4() {
        Volume f = NdimensionalBallVolumes.getVolume(4);
        
        assertEquals(1, f.numerator);
        assertEquals(2, f.denominator);
        assertEquals(2, f.pis);
        String s = f.toString();
        System.out.println(s);
    }
}