Complete the sequence generated by a polynomial

Difficulty Level : Hard
Last Updated : 27 Jul, 2021

Given a sequence with some of its term, we need to calculate next K term of this sequence. It is given that sequence is generated by some polynomial, however complex that polynomial can be. Notice polynomial is an expression of the following form:
P(x) = a0 + a1 x +a2 x^2 + a3 x^3 …….. + an x^n
The given sequence can always be described by a number of polynomials, among these polynomial we need to find polynomial with lowest degree and generate next terms using this polynomial only.

Examples:

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If given sequence is 1, 2, 3, 4, 5 then its next term will be 6, 7, 8, etc
and this correspond to a trivial polynomial.
If given sequence is 1, 4, 7, 10 then its next term will be 13, 16, etc.

We can solve this problem using a technique called difference of differences method, which is derivable from lagrange polynomial.

The technique is simple, we take the difference between the consecutive terms, if difference are equal then we stop and build up next term of the sequence otherwise we again take the difference between these differences until they become constant.
The technique is explained in below diagram with an example, given sequence is 8, 11, 16, 23 and we are suppose to find next 3 terms of this sequence.

generate-polynomial

In below, code same technique is implemented, first we loop until we get a constant difference keeping first number of each difference sequence in a separate vector for rebuilding the sequence again. Then we add K instance of same constant difference to our array for generating new K term of sequence and we follow same procedure in reverse order to rebuild the sequence.

See the below code for a better understanding.

C++

// C++ code to generate next terms of a given polynomial
// sequence
#include <bits/stdc++.h>
using namespace std;
 
//  method to print next terms term of sequence
void nextTermsInSequence(int sequence[], int N, int terms)
{
    int diff[N + terms];
 
    //  first copy the sequence itself into diff array
    for (int i = 0; i < N; i++)
        diff[i] = sequence[i];
 
    bool more = false;
    vector<int> first;
    int len = N;
 
    // loop until one difference remains or all
    // difference become constant
    while (len > 1)
    {
        // keeping the first term of sequence for
        // later rebuilding
        first.push_back(diff[0]);
        len--;
 
        // converting the difference to difference
        // of differences
        for (int i = 0; i < len; i++)
            diff[i] = diff[i + 1] - diff[i];
 
        // checking if all difference values are
        // same or not
        int i;
        for (i = 1; i < len; i++)
            if (diff[i] != diff[i - 1])
                break;
 
        // If some difference values were not same
        if (i != len)
           break;
    }
 
    int iteration = N - len;
 
    //  padding terms instance of constant difference
    // at the end of array
    for (int i = len; i < len + terms; i++)
        diff[i] = diff[i - 1];
    len += terms;
 
    //  iterating to get actual sequence back
    for (int i = 0; i < iteration; i++)
    {
        len++;
 
        //  shifting all difference by one place
        for (int j = len - 1; j > 0; j--)
            diff[j] = diff[j - 1];
 
        // copying actual first element
        diff[0] = first[first.size() - i - 1];
 
        // converting difference of differences to
        // difference array
        for (int j = 1; j < len; j++)
            diff[j] = diff[j - 1] + diff[j];
    }
 
    //  printing the result
    for (int i = 0; i < len; i++)
        cout << diff[i] << " ";
    cout << endl;
}
 
//  Driver code to test above method
int main()
{
    int sequence[] = {8, 11, 16, 23};
    int N = sizeof(sequence) / sizeof(int);
 
    int terms = 3;
    nextTermsInSequence(sequence, N, terms);
 
    return 0;
}

Java

// Java code to generate next terms
// of a given polynomial sequence
import java.util.*;
 
class GFG{
   
// Method to print next terms term of sequence
static void nextTermsInSequence(int []sequence,
                                int N, int terms)
{
    int []diff = new int[N + terms];
   
    // First copy the sequence itself
    // into diff array
    for(int i = 0; i < N; i++)
        diff[i] = sequence[i];
   
    //bool more = false;
    ArrayList<Object> first = new ArrayList<Object>();
    int len = N;
      
    // Loop until one difference remains
    // or all difference become constant
    while (len > 1)
    {
          
        // Keeping the first term of
        // sequence for later rebuilding
        first.add(diff[0]);
        len--;
   
        // Converting the difference to
        // difference of differences
        for(int i = 0; i < len; i++)
            diff[i] = diff[i + 1] - diff[i];
   
        // Checking if all difference values
        // are same or not
        int j;
        for(j = 1; j < len; j++)
            if (diff[j] != diff[j - 1])
                break;
   
        // If some difference values
        // were not same
        if (j != len)
           break;
    }
   
    int iteration = N - len;
   
    // Padding terms instance of constant
    // difference at the end of array
    for(int i = len; i < len + terms; i++)
        diff[i] = diff[i - 1];
          
    len += terms;
   
    // Iterating to get actual sequence back
    for(int i = 0; i < iteration; i++)
    {
        len++;
   
        // Shifting all difference by one place
        for(int j = len - 1; j > 0; j--)
            diff[j] = diff[j - 1];
   
        // Copying actual first element
        diff[0] = (int)first.get(first.size() - i - 1);
   
        // Converting difference of differences
        // to difference array
        for(int j = 1; j < len; j++)
            diff[j] = diff[j - 1] + diff[j];
    }
   
    // Printing the result
    for(int i = 0; i < len; i++)
    {
        System.out.print(diff[i] + " ");
    }
      
    System.out.println();
}
  
// Driver Code
public static void main(String[] args)
{
    int []sequence = { 8, 11, 16, 23 };
    int N = sequence.length;
    int terms = 3;
      
    nextTermsInSequence(sequence, N, terms);
}
}
  
// This code is contributed by pratham76

Python3

# Python3 code to generate next terms
# of a given polynomial sequence
 
# Method to print next terms term of sequence
def nextTermsInSequence(sequence, N, terms):
 
    diff = [0] * (N + terms)
 
    # First copy the sequence itself
    # into diff array
    for i in range(N):
        diff[i] = sequence[i]
 
    more = False
    first = []
    length = N
 
    # Loop until one difference remains
    # or all difference become constant
    while (length > 1):
     
        # Keeping the first term of sequence
        # for later rebuilding
        first.append(diff[0])
        length -= 1
 
        # Converting the difference to difference
        # of differences
        for i in range(length):
            diff[i] = diff[i + 1] - diff[i]
 
        # Checking if all difference values are
        # same or not
        for i in range(1, length):
            if (diff[i] != diff[i - 1]):
                break
 
        # If some difference values
        # were not same
        if (i != length):
            break
 
    iteration = N - length
 
    # Padding terms instance of constant
    # difference at the end of array
    for i in range(length, length + terms):
        diff[i] = diff[i - 1]
         
    length += terms
 
    # Iterating to get actual sequence back
    for i in range(iteration):
        length += 1
 
        # Shifting all difference by one place
        for j in range(length - 1, -1, -1):
            diff[j] = diff[j - 1]
 
        # Copying actual first element
        diff[0] = first[len(first) - i - 1]
 
        # Converting difference of differences to
        # difference array
        for j in range(1, length):
            diff[j] = diff[j - 1] + diff[j]
 
    # Printing the result
    for i in range(length):
        print(diff[i], end = " ")
         
    print ()
 
# Driver code
if __name__ == "__main__":
 
    sequence = [ 8, 11, 16, 23 ]
    N = len(sequence)
    terms = 3
     
    nextTermsInSequence(sequence, N, terms)
 
# This code is contributed by chitranayal

C#

// C# code to generate next terms
// of a given polynomial sequence
using System;
using System.Collections;
class GFG{
  
// Method to print next terms term of sequence
static void nextTermsInSequence(int []sequence,
                                int N, int terms)
{
    int []diff = new int[N + terms];
  
    // First copy the sequence itself
    // into diff array
    for(int i = 0; i < N; i++)
        diff[i] = sequence[i];
  
    //bool more = false;
    ArrayList first = new ArrayList();
    int len = N;
     
    // Loop until one difference remains
    // or all difference become constant
    while (len > 1)
    {
         
        // Keeping the first term of
        // sequence for later rebuilding
        first.Add(diff[0]);
        len--;
  
        // Converting the difference to
        // difference of differences
        for(int i = 0; i < len; i++)
            diff[i] = diff[i + 1] - diff[i];
  
        // Checking if all difference values
        // are same or not
        int j;
        for(j = 1; j < len; j++)
            if (diff[j] != diff[j - 1])
                break;
  
        // If some difference values
        // were not same
        if (j != len)
           break;
    }
  
    int iteration = N - len;
  
    // Padding terms instance of constant
    // difference at the end of array
    for(int i = len; i < len + terms; i++)
        diff[i] = diff[i - 1];
         
    len += terms;
  
    // Iterating to get actual sequence back
    for(int i = 0; i < iteration; i++)
    {
        len++;
  
        // Shifting all difference by one place
        for(int j = len - 1; j > 0; j--)
            diff[j] = diff[j - 1];
  
        // Copying actual first element
        diff[0] = (int)first[first.Count - i - 1];
  
        // Converting difference of differences
        // to difference array
        for(int j = 1; j < len; j++)
            diff[j] = diff[j - 1] + diff[j];
    }
  
    // Printing the result
    for(int i = 0; i < len; i++)
    {
        Console.Write(diff[i] + " ");
    }
     
    Console.Write("\n");
}
 
// Driver Code
public static void Main(string[] args)
{
    int []sequence = { 8, 11, 16, 23 };
    int N = sequence.Length;
    int terms = 3;
     
    nextTermsInSequence(sequence, N, terms);
}
}
 
// This code is contributed by rutvik_56

Javascript

<script>
 
// Javascript code to generate next terms
// of a given polynomial sequence
 
    // Method to print next terms term of sequence
    function nextTermsInSequence(sequence, N,terms)
    {
        let diff = new Array(N + terms);
        // First copy the sequence itself
        // into diff array
        for(let i = 0; i < N; i++)
            diff[i] = sequence[i];
        
        //bool more = false;
        let first=[];  
        let len = N;
           
        // Loop until one difference remains
        // or all difference become constant
        while (len > 1)
        {
               
            // Keeping the first term of
            // sequence for later rebuilding
            first.push(diff[0]);
            len--;
        
            // Converting the difference to
            // difference of differences
            for(let i = 0; i < len; i++)
                diff[i] = diff[i + 1] - diff[i];
        
            // Checking if all difference values
            // are same or not
            let j;
            for(j = 1; j < len; j++)
                if (diff[j] != diff[j - 1])
                    break;
        
            // If some difference values
            // were not same
            if (j != len)
               break;
        }
        let iteration = N - len;
    
        // Padding terms instance of constant
        // difference at the end of array
        for(let i = len; i < len + terms; i++)
            diff[i] = diff[i - 1];
               
        len += terms;
        
        // Iterating to get actual sequence back
        for(let i = 0; i < iteration; i++)
        {
            len++;
        
            // Shifting all difference by one place
            for(let j = len - 1; j > 0; j--)
                diff[j] = diff[j - 1];
        
            // Copying actual first element
            diff[0] = first[first.length - i - 1];
        
            // Converting difference of differences
            // to difference array
            for(let j = 1; j < len; j++)
                diff[j] = diff[j - 1] + diff[j];
        }
        
        // Printing the result
        for(let i = 0; i < len; i++)
        {
            document.write(diff[i] + " ");
        }
           
        document.write("<br>");
    }
     
    // Driver Code
    let sequence=[8, 11, 16, 23];
    let N = sequence.length;
    let terms = 3;
    nextTermsInSequence(sequence, N, terms);
 
     
    // This code is contributed by avanitrachhadiya2155
     
</script>

Output:

8 11 16 23 30 37 44

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Complete the sequence generated by a polynomial

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Python3

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Javascript