Dev: Added algorithm to find the nth number in the Sylvester Sequence (#6613)

Added algorithm to find the nth number in the SylvesterSequence

src/main/java/com/thealgorithms/recursion/SylvesterSequence.java

@@ -0,0 +1,50 @@
+package com.thealgorithms.recursion;
+
+import java.math.BigInteger;
+
+/**
+ * A utility class for calculating numbers in Sylvester's sequence.
+ *
+ * <p>Sylvester's sequence is a sequence of integers where each term is calculated
+ * using the formula:
+ * <pre>
+ * a(n) = a(n-1) * (a(n-1) - 1) + 1
+ * </pre>
+ * with the first term being 2.
+ *
+ * <p>This class is final and cannot be instantiated.
+ *
+ * @see <a href="https://en.wikipedia.org/wiki/Sylvester_sequence">Wikipedia: Sylvester sequence</a>
+ */
+public final class SylvesterSequence {
+
+    // Private constructor to prevent instantiation
+    private SylvesterSequence() {
+    }
+
+    /**
+     * Calculates the nth number in Sylvester's sequence.
+     *
+     * <p>The sequence is defined recursively, with the first term being 2:
+     * <pre>
+     * a(1) = 2
+     * a(n) = a(n-1) * (a(n-1) - 1) + 1 for n > 1
+     * </pre>
+     *
+     * @param n the position in the sequence (must be greater than 0)
+     * @return the nth number in Sylvester's sequence
+     * @throws IllegalArgumentException if n is less than or equal to 0
+     */
+    public static BigInteger sylvester(int n) {
+        if (n <= 0) {
+            throw new IllegalArgumentException("sylvester() does not accept negative numbers or zero.");
+        }
+        if (n == 1) {
+            return BigInteger.valueOf(2);
+        } else {
+            BigInteger prev = sylvester(n - 1);
+            // Sylvester sequence formula: a(n) = a(n-1) * (a(n-1) - 1) + 1
+            return prev.multiply(prev.subtract(BigInteger.ONE)).add(BigInteger.ONE);
+        }
+    }
+}

src/test/java/com/thealgorithms/recursion/SylvesterSequenceTest.java

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+package com.thealgorithms.recursion;
+
+import static org.junit.jupiter.api.Assertions.assertEquals;
+import static org.junit.jupiter.api.Assertions.assertNotNull;
+import static org.junit.jupiter.api.Assertions.assertThrows;
+import static org.junit.jupiter.api.Assertions.assertTrue;
+
+import java.math.BigInteger;
+import java.util.stream.Stream;
+import org.junit.jupiter.api.Test;
+import org.junit.jupiter.params.ParameterizedTest;
+import org.junit.jupiter.params.provider.MethodSource;
+import org.junit.jupiter.params.provider.ValueSource;
+
+class SylvesterSequenceTest {
+
+    /**
+     * Provides test cases for valid Sylvester sequence numbers.
+     * Format: { n, expectedValue }
+     */
+    static Stream<Object[]> validSylvesterNumbers() {
+        return Stream.of(new Object[] {1, BigInteger.valueOf(2)}, new Object[] {2, BigInteger.valueOf(3)}, new Object[] {3, BigInteger.valueOf(7)}, new Object[] {4, BigInteger.valueOf(43)}, new Object[] {5, BigInteger.valueOf(1807)}, new Object[] {6, new BigInteger("3263443")},
+            new Object[] {7, new BigInteger("10650056950807")}, new Object[] {8, new BigInteger("113423713055421844361000443")});
+    }
+
+    @ParameterizedTest
+    @MethodSource("validSylvesterNumbers")
+    void testSylvesterValidNumbers(int n, BigInteger expected) {
+        assertEquals(expected, SylvesterSequence.sylvester(n), "Sylvester sequence value mismatch for n=" + n);
+    }
+
+    /**
+     * Test edge case for n <= 0 which should throw IllegalArgumentException
+     */
+    @ParameterizedTest
+    @ValueSource(ints = {0, -1, -10, -100})
+    void testSylvesterInvalidZero(int n) {
+        assertThrows(IllegalArgumentException.class, () -> SylvesterSequence.sylvester(n));
+    }
+
+    /**
+     * Test a larger number to ensure no overflow occurs.
+     */
+    @Test
+    void testSylvesterLargeNumber() {
+        int n = 10;
+        BigInteger result = SylvesterSequence.sylvester(n);
+        assertNotNull(result);
+        assertTrue(result.compareTo(BigInteger.ZERO) > 0, "Result should be positive");
+    }
+}