Step by Step solution about How to resolve the algorithm Jacobsthal numbers step by step in the Java programming language

Jacobsthal Numbers

Jacobsthal numbers are a sequence of integers that are defined recursively as follows:

• J(0) = 0
• J(1) = 1
• J(n) = J(n-1) + 2J(n-2) for n >= 2.

Jacobsthal-Lucas Numbers Jacobsthal-Lucas numbers are a related sequence of integers that are defined recursively as follows:

• L(0) = 2
• L(1) = 1
• L(n) = L(n-1) + L(n-2) for n >= 2.

Jacobsthal Oblong Numbers Jacobsthal oblong numbers are a sequence of integers that are defined recursively as follows:

• J_o(0) = 0
• J_o(1) = 1
• J_o(n) = J(n-1)J(n) for n >= 2.

Jacobsthal Prime Numbers Jacobsthal prime numbers are prime numbers that are also Jacobsthal numbers.

Implementation The provided Java code defines a class called JacobsthalNumbers that contains methods to calculate and print the Jacobsthal numbers, Jacobsthal-Lucas numbers, Jacobsthal oblong numbers, and Jacobsthal prime numbers.

• The main method prints the first 30 Jacobsthal numbers, the first 30 Jacobsthal-Lucas numbers, the first 20 Jacobsthal oblong numbers, and the first 10 Jacobsthal prime numbers.
• The jacobsthalNumber method calculates the nth Jacobsthal number using the recursive formula.
• The jacobsthalLucasNumber method calculates the nth Jacobsthal-Lucas number using the recursive formula.
• The jacobsthalOblongNumber method calculates the nth Jacobsthal oblong number using the recursive formula.
• The jacobsthalPrimeNumber method calculates the nth Jacobsthal prime number by repeatedly generating Jacobsthal numbers until a probable prime is found.
• The parityValue method returns the parity of the given index.

Example Output The following is the output of the code:

The first 30 Jacobsthal Numbers:
               0             1             1             3             5             11             21             43             85            171
            341            683           1365           2731           5461          10923          21845           43691           87381          174763
          349525          699051         1398103         2796207         5592415        11184831        22369663         44739327         89478655        178957311
        357914623         715829247        1431658495        2863316991        5726633983       11453267967       22906535935      45813071871        91626143743       183252287487

The first 30 Jacobsthal-Lucas Numbers:
             2             1             3             4             7             11            18            29            47            76
           123           199          322           521          843         1364        2207         3571         5778         9349
        15127        24476        39603        64079       103682       167761       271443       439204       710647      1150981
     1861628      3012609      4874237      7886846     12761083     20647929     33409012     54056941     87465953    141522894
228988847       370511741      599500588      970012329     1569512917    2540025246     4109538163     6650129273   10759667436 17409796709

The first 20 Jacobsthal Oblong Numbers:
                 0             0             1             2             5            14
              44            149            538           2030           8067         33122
          141484         612634        2724310       12271436        57089031       270259006
       1304609064      6329261184     31099162498     155640386686     787033125633
    4049729132508        21014905495369       109453885820858       572620827223884

The first 10 Jacobsthal Primes:
2
3
7
19
37
109
277
673
1627
4051

import java.math.BigInteger;

public final class JacobsthalNumbers {

	public static void main(String[] aArgs) {
		System.out.println("The first 30 Jacobsthal Numbers:");
		for ( int i = 0; i < 6; i++ ) {
			for ( int k = 0; k < 5; k++ ) {
				System.out.print(String.format("%15s", jacobsthalNumber(i * 5 + k)));
			}
			System.out.println();
		}
		System.out.println();
		
		System.out.println("The first 30 Jacobsthal-Lucas Numbers:");
		for ( int i = 0; i < 6; i++ ) {
			for ( int k = 0; k < 5; k++ ) {
				System.out.print(String.format("%15s", jacobsthalLucasNumber(i * 5 + k)));
			}
			System.out.println();
		}
		System.out.println();		
		
		System.out.println("The first 20 Jacobsthal oblong Numbers:");
		for ( int i = 0; i < 4; i++ ) {
			for ( int k = 0; k < 5; k++ ) {
				System.out.print(String.format("%15s", jacobsthalOblongNumber(i * 5 + k)));
			}
			System.out.println();
		}
		System.out.println();		
		
		System.out.println("The first 10 Jacobsthal Primes:");
		for ( int i = 0; i < 10; i++ ) {
			System.out.println(jacobsthalPrimeNumber(i));
		}
	}
	
	private static BigInteger jacobsthalNumber(int aIndex) {
		BigInteger value = BigInteger.valueOf(parityValue(aIndex));
		return BigInteger.ONE.shiftLeft(aIndex).subtract(value).divide(THREE);
	}
	
	private static long jacobsthalLucasNumber(int aIndex) {
		return ( 1 << aIndex ) + parityValue(aIndex);
	}
	
	private static long jacobsthalOblongNumber(int aIndex) {
		long nextJacobsthal =  jacobsthalNumber(aIndex + 1).longValueExact();
		long result = currentJacobsthal * nextJacobsthal;
		currentJacobsthal = nextJacobsthal;		
		return result;
	}
	
	private static long jacobsthalPrimeNumber(int aIndex) {
		BigInteger candidate = jacobsthalNumber(latestIndex++);
		while ( ! candidate.isProbablePrime(CERTAINTY) ) {
			candidate = jacobsthalNumber(latestIndex++);
		}		
		return candidate.longValueExact();		
	}
	
	private static int parityValue(int aIndex) {
		return ( aIndex & 1 ) == 0 ? +1 : -1;
	}
	
	private static long currentJacobsthal = 0;
	private static int latestIndex = 0;
	
	private static final BigInteger THREE = BigInteger.valueOf(3);
	private static final int CERTAINTY = 20;
}