# Complete the sequence generated by a polynomial

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:

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++

`// 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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