# Connell Sequence

Given an integer ‘n’, generate the first ‘n’ terms of the Connell Sequence.
Connell Sequence is the sequence formed with the first odd number, i.e 1 as its first term. The subsequent terms of the sequence are made up of the first two even numbers, i.e 2 and 4, followed by the next three odd numbers, i.e 5, 7 and 9, followed by the next four even numbers, i.e 10, 12, 14 and 16 and so on …. the sequence continues.

Formula:

```a[n] = 2 * n - floor((1 + sqrt(8 * n - 7))/2)   ; n > 1
```

Examples:

```Input : 6
Output : 1 2 4 5 7 9

Input : 12
Output : 1 2 4 5 7 9 10 12 14 16 17 19
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

It may be noted here that writing the terms in new lines as, first term in first line, next two terms in next line, next three terms in next line and so on, gives an interesting pattern as:

Line 1 : 1
Line 2 : 2 4
Line 3 : 5 7 9
Line 4 : 10 12 14 16
Line 5 : 17 19 21 23 25
and so on…
The pattern is every last number of a particular line is equal to that line number squared.
For example

1. In line 2 last number is 4 which is equal to its line number squared, i.e 2^2
2. In line 5 last number is 25 which is equal to its line number squared, i.e 5^2

Below is a simple implementation where we generate result by alternatively adding odd and even number of elements. We use size of current list to decide next number of elements to push.

## C++

 `// CPP code to generate first 'n' terms ` `// of Connell Sequence ` `#include ` `using` `namespace` `std; ` ` `  `// Function to generate a fixed number ` `// of even or odd terms. The size of r ` `// decides whether numbers to be generated ` `// even or odd. ` `vector<``long` `long` `int``> gen(``long` `long` `int` `n, ` `                  ``vector<``long` `long` `int``> r) ` `{ ` `    ``long` `long` `int` `a = r[r.size() - 1]; ` `    ``a++; ` `    ``for` `(``int` `i = 1; i <= n; a += 2, i++) ` `        ``r.push_back(a); ` `    ``return` `r; ` `} ` ` `  `// Generating the first 'n' terms of ` `// Connell Sequence ` `vector<``long` `long` `int``> conell(``long` `long` `int` `n) ` `{ ` `    ``vector<``long` `long` `int``> res; ` `    ``long` `long` `int` `k = 1; ` ` `  `    ``// A dummy 0 is inserted at the ` `    ``// beginning for consistency ` `    ``res.push_back(0); ` ` `  `    ``while` `(1) ` `    ``{ ` `        ``// Calling function gen() to generate ` `        ``// 'k' number of terms ` `        ``res = gen(k, res); ` `        ``k++; ` ` `  `        ``int` `j = res.size() - 1; ` `        ``while` `(j != n && j + k > n) ` `            ``k--; ` ` `  `        ``// Checking if 'n' terms are ` `        ``// already generated ` `        ``if` `(j >= n) ` `            ``break``; ` `    ``} ` ` `  `    ``// Removing the previously inserted dummy 0 ` `    ``res.erase(res.begin()); ` ` `  `    ``return` `res; ` `} ` ` `  `// Driver Method ` `int` `main() ` `{ ` `    ``long` `long` `int` `n = 10; ` ` `  `    ``cout << ``"The first "` `<< n ` `         ``<< ``" terms are"` `<< endl; ` `    ``vector<``long` `long` `int``> res = conell(n); ` `    ``for` `(``int` `i = 0; i < res.size(); i++) ` `        ``cout << res[i] << ``" "``; ` `    ``cout << endl; ` `    ``return` `0; ` `} `

## Java

 `// Java code to generate  ` `// first 'n' terms  ` `// of Connell Sequence  ` `import` `java.util.*; ` ` `  `class` `GFG  ` `{ ` `     `  `    ``// Function to generate a  ` `    ``// fixed number of even or  ` `    ``// odd terms. The size of r  ` `    ``// decides whether numbers  ` `    ``// to be generated even or odd.  ` ` `  `    ``static` `Vector gen(``long` `n, Vector r)  ` `    ``{      ` `        ``long` `a = r.get(r.size() - ``1``);      ` `        ``a++;      ` `        ``for` `(``int` `i = ``1``; i <= n; a += ``2``, i++)  ` `        ``{ ` `            ``r.add(a); ` `        ``}      ` `        ``return` `r;      ` `    ``} ` ` `  `    ``// Generating the first  ` `    ``// 'n' terms of  ` `    ``// Connell Sequence  ` `    ``static` `Vector conell(``long` `n)  ` `    ``{      ` `        ``Vector res = ``new` `Vector();      ` `        ``long` `k = ``1``; ` ` `  `        ``// A dummy 0 is inserted  ` `        ``// at the beginning for  ` `        ``// consistency  ` `        ``res.add(0L);      ` `         `  `        ``while` `(``true``) ` `        ``{ ` `            ``// Calling function  ` `            ``// gen() to generate  ` `            ``// 'k' number of terms  ` `            ``res = gen(k, res);          ` `            ``k++;          ` `             `  `            ``int` `j = res.size() - ``1``;          ` `            ``while` `(j != n && j + k > n) ` `            ``{ ` `                ``k--; ` `            ``} ` ` `  `            ``// Checking if 'n'  ` `            ``// terms are already  ` `            ``// generated  ` `            ``if` `(j >= n)  ` `            ``{ ` `                ``break``; ` `            ``}          ` `        ``} ` ` `  `        ``// Removing the previously  ` `        ``// inserted dummy 0  ` `        ``res.remove(``0``);      ` `         `  `        ``return` `res;      ` `    ``} ` ` `  `    ``// Driver Code  ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``long` `n = ``10``;      ` `         `  `        ``System.out.println(``"The first "` `                    ``+ n + ``" terms are"``); ` `                 `  `        ``Vector res = conell(n);      ` `        ``for` `(``int` `i = ``0``; i < res.size(); i++) ` `        ``{ ` `            ``System.out.print(res.get(i) + ``" "``); ` `        ``}      ` `        ``System.out.println();      ` `    ``}  ` `}  ` ` `  `// This code has been contributed ` `// by Rajput-Ji `

## Python3

 `# Python3 code to generate first 'n' terms ` `# of Connell Sequence ` ` `  `# Function to generate a fixed number ` `# of even or odd terms. The size of r ` `# decides whether numbers to be generated ` `# even or odd. ` `def` `gen(n, r): ` `    ``a ``=` `r[``-``1``] ` `    ``a ``+``=` `1` `    ``for` `i ``in` `range``(``1``, n ``+` `1``): ` `        ``r.append(a) ` `        ``a ``+``=` `2` `    ``return` `r ` ` `  `# Generating the first 'n' terms of ` `# Connell Sequence ` `def` `conell(n): ` `    ``res ``=` `[] ` `    ``k ``=` `1` ` `  `    ``# A dummy 0 is inserted at the ` `    ``# beginning for consistency ` `    ``res.append(``0``) ` ` `  `    ``while` `1``: ` ` `  `        ``# Calling function gen() to generate ` `        ``# 'k' number of terms ` `        ``res ``=` `gen(k, res) ` `        ``k ``+``=` `1` `        ``j ``=` `len``(res) ``-` `1` `        ``while` `j !``=` `n ``and` `j ``+` `k > n: ` `            ``k ``-``=` `1` ` `  `        ``# Checking if 'n' terms are ` `        ``# already generated ` `        ``if` `j >``=` `n: ` `            ``break` ` `  `    ``# Removing the previously inserted dummy 0 ` `    ``res.remove(res[``0``]) ` `    ``return` `res ` ` `  `# Driver Code ` `if` `__name__ ``=``=` `"__main__"``: ` `    ``n ``=` `10` `    ``print``(``"The first %d terms are"` `%` `n) ` `    ``res ``=` `conell(n) ` `    ``for` `i ``in` `range``(``len``(res)): ` `        ``print``(res[i], end ``=` `" "``) ` `    ``print``() ` ` `  `# This code is contributed by ` `# sanjeev2552 `

## C#

 `// C# code to generate ` `// first 'n' terms ` `// of Connell Sequence ` `using` `System; ` `using` `System.Collections.Generic; ` ` `  `class` `GFG ` `{ ` `    ``// Function to generate a  ` `    ``// fixed number of even or  ` `    ``// odd terms. The size of r ` `    ``// decides whether numbers  ` `    ``// to be generated even or odd. ` `    ``static` `List<``long``> gen(``long` `n,  ` `                          ``List<``long``> r) ` `    ``{ ` `        ``long` `a = r[r.Count - 1]; ` `        ``a++; ` `        ``for` `(``int` `i = 1; i <= n;  ` `                 ``a += 2, i++) ` `            ``r.Add(a); ` `        ``return` `r; ` `    ``} ` `     `  `    ``// Generating the first ` `    ``// 'n' terms of ` `    ``// Connell Sequence ` `    ``static` `List<``long``> conell(``long` `n) ` `    ``{ ` `        ``List<``long``> res = ``new` `List<``long``>(); ` `        ``long` `k = 1; ` `     `  `        ``// A dummy 0 is inserted  ` `        ``// at the beginning for ` `        ``// consistency ` `        ``res.Add(0); ` `     `  `        ``while` `(``true``) ` `        ``{ ` `            ``// Calling function  ` `            ``// gen() to generate ` `            ``// 'k' number of terms ` `            ``res = gen(k, res); ` `            ``k++; ` `     `  `            ``int` `j = res.Count - 1; ` `            ``while` `(j != n &&  ` `                   ``j + k > n) ` `                ``k--; ` `     `  `            ``// Checking if 'n'  ` `            ``// terms are already  ` `            ``// generated ` `            ``if` `(j >= n) ` `                ``break``; ` `        ``} ` `     `  `        ``// Removing the previously  ` `        ``// inserted dummy 0 ` `        ``res.RemoveAt(0); ` `     `  `        ``return` `res; ` `    ``} ` `     `  `    ``// Driver Code ` `    ``static` `void` `Main() ` `    ``{ ` `        ``long` `n = 10; ` `     `  `        ``Console.WriteLine(``"The first "` `+  ` `                      ``n + ``" terms are"``); ` `        ``List<``long``> res = conell(n); ` `        ``for` `(``int` `i = 0; i < res.Count; i++) ` `            ``Console.Write(res[i] + ``" "``); ` `        ``Console.WriteLine(); ` `    ``} ` `} ` ` `  `// This code is contributed by  ` `// Manish Shaw(manishshaw1) `

Output:

```The first 10 terms are
1 2 4 5 7 9 10 12 14 16
```

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