N Bonacci Numbers In C++

Think of the N-bonacci sequence as a relay race where each runner hands off his speed to the next N runners, so there is a growth chain reaction. The Fibonacci sequence can be interestingly extended to the N-bonacci numbers. The sum of the two terms that come before is utilized in the Fibonacci sequence, whereas the sum of the final N terms is utilized in the N-bonacci sequence.

This generalization opens up various possibilities for computer implementation and mathematical inquiry. The definition of N-bonacci numbers, their characteristics, and a practical C++ implementation will all be covered in this article.

What are N-bonacci Numbers?

The N-bonacci sequence is a generalization of the Fibonacci sequence, where instead of summing the previous two terms, we sum the previous N terms to get the next term.

For example:

• ✅ In the Tribonnaci sequence (N=3), each one term is the sum of the last three terms.
• ✅ In the Quadbonacci sequence (N=4), each one term is the sum of the last four terms.

Properties of N-bonacci Numbers:

Several properties of N-bonacci Numbers in C++ are as follows:

• ✅ Growth Rate: Like Fibonacci numbers, N-bonacci numbers grow exponentially. However, the growth rate increases with N, as more preceding terms affect each one term.
• ✅ Initial Conditions: For N-bonacci numbers, the first N terms are assigned specific initial conditions, typically zeros followed by one. It ensures that the sequence is uniquely defined.
• ✅ Generalized Golden Ratio: The successive N-bonacci numbers' ratio approximates the generalized golden ratio based on N.
• ✅ Applications: From cryptography to dynamic algorithms, N-bonacci numbers can model complex systems that depend on extended memory that require extended memory of past states.

Implementing N-bonacci Numbers in C++

The best approach for properly computing N-bonacci number is to closely track memory utilization and use smart loops.

Step 1: Setting Up the Function

First, we start by writing a function that takes N and the desired term num as input and outputs the N-bonacci sequence up to the num th term.

#include <iostream>
#include <vector>
using namespace std;
void generateNBonacci(int N, int num) 
{
    // Edge case handling
    if (num <= 0) 
{
        cout << "Invalid input. Number of terms should be greater than 0." << endl;
        return;
    }
    // Initialize the N-bonacci sequence
    vector<long long> sequence(num, 0);
    if (num > N) 
{
        sequence[N - 1] = 1;
    }
 else 
{
        sequence[num - 1] = 1;
    }
    // Generate N-bonacci sequence
    for (int i = N; i < num; ++i) 
{
        for (int j = i - 1; j >= i - N; --j) 
{
            sequence[i] += sequence[j];
        }
    }
    // Print the sequence
    for (const auto &val : sequence) 
{
        cout << val << " ";
    }
    cout << endl;
}

Step 2: Main Function

The main function serves as the entry point to the program, where the user can input values for N and num.

int main() 
{
    int N, num;
    cout << "Enter the value of N (order of the sequence): ";
    cin >> N;
    cout << "Enter the number of terms to generate: ";
    cin >> num;
    generateNBonacci(N, num);
    return 0;
}

Optimizations for Large Values:

1. Sliding Window Technique:

Instead of recalculating the sum of the last N terms repeatedly, use a sliding window approach to maintain the sum dynamically.

void generateNBonacciOptimized(int N, int num) 
{
    if (num <= 0) 
{
        cout << "Invalid input. Number of terms should be greater than 0." << endl;
        return;
    }
    vector<long long> sequence(num, 0);
    if (num > N) 
{
        sequence[N - 1] = 1;
    }
 else 
{
        sequence[num - 1] = 1;
    }
    long long windowSum = 1; // Sum of the first N terms (1 at index N-1)
    for (int i = N; i < num; ++i) 
{
        sequence[i] = windowSum;
        windowSum += sequence[i] - sequence[i - N];
    }
    for (const auto &val : sequence) 
{
        cout << val << " ";
    }
    cout << endl;
}

1. Modular Arithmetic:

For extremely large values, consider computing the sequence modulo a prime number to avoid overflow.

1. Matrix Exponentiation:

Use matrix exponentiation techniques to represent the recurrence relation for calculating specific terms in constant time.

Example Outputs:

Output:

Input:
N = 3, num = 10
Output:
0 0 1 1 2 4 7 13 24 44
Input:
N = 4, num = 10
Output:
0 0 0 1 1 2 4 8 15 29

Applications in Real-World Scenarios:

Several applications of N-bonacci Numbers in C++ are as follows:

1. Modeling Systems with Extended Memory:

These N-bonacci numbers describe systems for which the current state depends on a greater history of previous states.

2. Algorithm Design

Dynamic programming problems and cryptographic hash functions.

3. Education

A useful example in teaching about recursion series and techniques of efficient computation.