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# Program to find Nth term of the Van Eck’s Sequence

Given a positive integer N, the task is to print Nth term of the Van Eck’s sequence.
In mathematics, Van Eck’s sequence is an integer sequence which is defined recursively as follows:

• Let the first term be 0 i.e a0 = 0.
• Then for n >= 0, if there exists an m < n such that
`am = an`
• take the largest such m and set an+1 = n − m;
• Otherwise an+1 = 0.

First few terms of Van Eck’s Sequence are as follows:

0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 0, 2, 6, 5, 4, 0, 5 …

Example:

```Input: N = 5
Output: 2

Input: N = 10
Output: 6 ```

Approach:
As described above we can follow the below steps to generate Van Eck’s sequence:

• Set the first term of the sequence as 0.
• Then Repeatedly apply:
• If the last term has not occurred yet and is new to the sequence so far then, set the next term as zero.
• Otherwise, the next term is how far back this last term has occurred previously.
• Once the sequence is generated we can get our nth term easily.

Below is the implementation of above approach:

## C++

 `// C++ program to print Nth``// term of Van Eck's sequence` `#include ``using` `namespace` `std;` `#define MAX 1000``int` `sequence[MAX];` `// Utility function to compute``// Van Eck's sequence``void` `vanEckSequence()``{` `    ``// Initialize sequence array``    ``for` `(``int` `i = 0; i < MAX; i++) {``        ``sequence[i] = 0;``    ``}` `    ``// Loop to generate sequence``    ``for` `(``int` `i = 0; i < MAX; i++) {` `        ``// Check if sequence[i] has occurred``        ``// previously or is new to sequence``        ``for` `(``int` `j = i - 1; j >= 0; j--) {``            ``if` `(sequence[j] == sequence[i]) {` `                ``// If occurrence found``                ``// then the next term will be``                ``// how far back this last term``                ``// occurred previously``                ``sequence[i + 1] = i - j;``                ``break``;``            ``}``        ``}``    ``}``}` `// Utility function to return``// Nth term of sequence``int` `getNthTerm(``int` `n)``{` `    ``return` `sequence[n];``}` `// Driver code``int` `main()``{` `    ``// Pre-compute Van Eck's sequence``    ``vanEckSequence();` `    ``int` `n = 6;` `    ``// Print nth term of the sequence``    ``cout << getNthTerm(n) << endl;` `    ``n = 100;` `    ``// Print nth term of the sequence``    ``cout << getNthTerm(n) << endl;` `    ``return` `0;``}`

## Java

 `// Java program to print Nth``// term of Van Eck's sequence` `class` `GFG {` `    ``static` `int` `MAX = ``1000``;` `    ``// Array to store terms of sequence``    ``static` `int` `sequence[] = ``new` `int``[MAX];` `    ``// Utility function to compute``    ``// Van Eck's sequence``    ``static` `void` `vanEckSequence()``    ``{` `        ``// Initialize sequence array``        ``for` `(``int` `i = ``0``; i < MAX - ``1``; i++) {``            ``sequence[i] = ``0``;``        ``}` `        ``// Loop to generate sequence``        ``for` `(``int` `i = ``0``; i < MAX - ``1``; i++) {` `            ``// Check if sequence[i] has occurred``            ``// previously or is new to sequence``            ``for` `(``int` `j = i - ``1``; j >= ``0``; j--) {``                ``if` `(sequence[j] == sequence[i]) {` `                    ``// If occurrence found``                    ``// then the next term will be``                    ``// how far back this last term``                    ``// occurred previously``                    ``sequence[i + ``1``] = i - j;``                    ``break``;``                ``}``            ``}``        ``}``    ``}` `    ``// Utility function to return``    ``// Nth term of sequence``    ``static` `int` `getNthTerm(``int` `n)``    ``{` `        ``return` `sequence[n];``    ``}` `    ``// Driver code``    ``public` `static` `void` `main(String[] args)``    ``{` `        ``// Pre-compute Van Eck's sequence``        ``vanEckSequence();` `        ``int` `n = ``6``;` `        ``// Print nth term of the sequence``        ``System.out.println(getNthTerm(n));` `        ``n = ``100``;` `        ``// Print nth term of the sequence``        ``System.out.println(getNthTerm(n));``    ``}``}`

## Python3

 `# Python3 program to print Nth``# term of Van Eck's sequence``MAX` `=` `1000``sequence ``=` `[``0``] ``*` `(``MAX` `+` `1``);` `# Utility function to compute``# Van Eck's sequence``def` `vanEckSequence() :` `    ``# Initialize sequence array``    ``for` `i ``in` `range``(``MAX``) :``        ``sequence[i] ``=` `0``;` `    ``# Loop to generate sequence``    ``for` `i ``in` `range``(``MAX``) :``        ` `        ``# Check if sequence[i] has occurred``        ``# previously or is new to sequence``        ``for` `j ``in` `range``(i ``-` `1` `, ``-``1``, ``-``1``) :``            ``if` `(sequence[j] ``=``=` `sequence[i]) :` `                ``# If occurrence found``                ``# then the next term will be``                ``# how far back this last term``                ``# occurred previously``                ``sequence[i ``+` `1``] ``=` `i ``-` `j;``                ``break``;` `# Utility function to return``# Nth term of sequence``def` `getNthTerm(n) :` `    ``return` `sequence[n];` `# Driver code``if` `__name__ ``=``=` `"__main__"` `:` `    ``# Pre-compute Van Eck's sequence``    ``vanEckSequence();` `    ``n ``=` `6``;` `    ``# Print nth term of the sequence``    ``print``(getNthTerm(n));` `    ``n ``=` `100``;` `    ``# Print nth term of the sequence``    ``print``(getNthTerm(n));` `# This code is contributed by kanugargng`

## C#

 `// C# program to print Nth``// term of Van Eck's sequence` `using` `System;``class` `GFG {` `    ``static` `int` `MAX = 1000;` `    ``// Array to store terms of sequence``    ``static` `int``[] sequence = ``new` `int``[MAX];` `    ``// Utility function to compute``    ``// Van Eck's sequence``    ``static` `void` `vanEckSequence()``    ``{` `        ``// Initialize sequence array``        ``for` `(``int` `i = 0; i < MAX - 1; i++) {``            ``sequence[i] = 0;``        ``}` `        ``// Loop to generate sequence``        ``for` `(``int` `i = 0; i < MAX - 1; i++) {` `            ``// Check if sequence[i] has occurred``            ``// previously or is new to sequence``            ``for` `(``int` `j = i - 1; j >= 0; j--) {``                ``if` `(sequence[j] == sequence[i]) {` `                    ``// If occurrence found``                    ``// then the next term will be``                    ``// how far back this last term``                    ``// occurred previously``                    ``sequence[i + 1] = i - j;``                    ``break``;``                ``}``            ``}``        ``}``    ``}` `    ``// Utility function to return``    ``// Nth term of sequence``    ``static` `int` `getNthTerm(``int` `n)``    ``{` `        ``return` `sequence[n];``    ``}` `    ``// Driver code``    ``public` `static` `void` `Main()``    ``{` `        ``// Pre-compute Van Eck's sequence``        ``vanEckSequence();` `        ``int` `n = 6;` `        ``// Print nth term of the sequence``        ``Console.WriteLine(getNthTerm(n));` `        ``n = 100;` `        ``// Print nth term of the sequence``        ``Console.WriteLine(getNthTerm(n));``    ``}``}`

## Javascript

 ``

Output:

```2
23```

Time Complexity: O(MAX2)
Auxiliary Space: O(MAX)

Method 2: Using lambda functions
There is no need to store the entire sequence to get the value of nth term. We can recursively build the series upto nth term and find only the value of nth term using the lambda expression.
Below is the implementation of the above approach using lambda function:

## Python3

 `# Python3 program to find``# Nth term of Van Eck's sequence``# using the lambda expression` `# Lambda function``f ``=` `lambda` `n, l ``=` `0``, ``*``s: f(n``-``1``, l ``in` `s ``and` `~s.index(l), l, ``*``s) \``    ``if` `n ``else` `-``l` `# The above lambda function recursively``# build the sequence and store it in tuple s` `# The expression l in s and ~s.index(l)``# returns False if l is not present in tuple s``# otherwise, returns the negation of the value of``# the index of l in tuple s``# and is appended to tuple s``# Thus, tuple s store negation of all the elements``# of the sequence in reverse order` `# At the end, when n reaches 0, function converts``# the nth term back to its actual value``# and returns it.`  `# Driver code``n ``=` `6` `# Get Nth term of the sequence``print``(f(n))` `n ``=` `100``# Get Nth term of the sequence``print``(f(n))`

## Javascript

 `// Lambda function``const f = (n, l = 0, ...s) =>``  ``// Recursively build the sequence and store it in array s``  ``n ? f(n-1, s.includes(l) ? ~s.indexOf(l) : 0, l, ...s) :``  ``// Convert the nth term back to its actual value and return it``  ``-l;` `// The expression s.includes(l) ? ~s.indexOf(l) : 0``// returns 0 if l is not present in array s``// otherwise, returns the negation of the value of``// the index of l in array s``// and is appended to array s``// Thus, array s store negation of all the elements``// of the sequence in reverse order` `// Driver code``let x= 6;` `// Get Nth term of the sequence``console.log(f(x));` `x = 100;` `// Get Nth term of the sequence``console.log(f(x));`

Output:

```2
23```

Time Complexity: O(n)
Auxiliary Space: O(n)

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