Complete the sequence generated by a polynomial

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^nThe 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:
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.
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++
Java
Python3
C#
Javascript
// 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 sequencevoid 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 methodint main(){    int sequence[] = {8, 11, 16, 23};    int N = sizeof(sequence) / sizeof(int);     int terms = 3;    nextTermsInSequence(sequence, N, terms);     return 0;}
// Java code to generate next terms// of a given polynomial sequenceimport java.util.*; class GFG{   // Method to print next terms term of sequencestatic 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 Codepublic 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 code to generate next terms# of a given polynomial sequence # Method to print next terms term of sequencedef 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 codeif __name__ == "__main__":     sequence = [ 8, 11, 16, 23 ]    N = len(sequence)    terms = 3         nextTermsInSequence(sequence, N, terms) # This code is contributed by chitranayal
// C# code to generate next terms// of a given polynomial sequenceusing System;using System.Collections;class GFG{  // Method to print next terms term of sequencestatic 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 Codepublic 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
<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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