# Number of GP (Geometric Progression) subsequences of size 3

Given n elements and a ratio r, find the number of G.P. subsequences with length 3. A subsequence is considered GP with length 3 with ration r.

Examples:

```Input : arr[] = {1, 1, 2, 2, 4}
r = 2
Output : 4
Explanation: Any of the two 1s can be chosen
as the first element, the second element can
be any of the two 2s, and the third element
of the subsequence must be equal to 4.

Input : arr[] = {1, 1, 2, 2, 4}
r = 3
Output : 0
```

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

A naive approach is to use three nested for loops and check for every subsequence with length 3 and keep a count of the subsequences. The complexity is O(n3).

An efficient approach is to solve the problem for fixed middle element of progression. This means that if we fix element a[i] as middle, then it must be multiple of r, and a[i]/r and a[i]*r must be present. We count number of occurrences of a[i]/r and a[i]*r and then multiply the counts. To do this, we can use concept of hashing where we store the count of all possible elements in two hash maps, one indicating the number of elements in the left and the other indicating the number of elements to the right.

Below is the implementation of the above approach

## C++

 `// CPP program to count GP subsequences of size 3. ` `#include ` `using` `namespace` `std; ` ` `  `// Returns count of G.P. subseqeunces ` `// with length 3 and common ratio r ` `long` `long` `subsequences(``int` `a[], ``int` `n, ``int` `r) ` `{ ` `    ``// hashing to maintain left and right array ` `    ``// elements to the main count ` `    ``unordered_map<``int``, ``int``> left, right; ` ` `  `    ``// stores the answer ` `    ``long` `long` `ans = 0; ` ` `  `    ``// traverse through the elements ` `    ``for` `(``int` `i = 0; i < n; i++) ` `        ``right[a[i]]++; ``// keep the count in the hash ` ` `  `    ``// traverse through all elements ` `    ``// and find out the number of elements as k1*k2 ` `    ``for` `(``int` `i = 0; i < n; i++) { ` ` `  `        ``// keep the count of left and right elements ` `        ``// left is a[i]/r and right a[i]*r ` `        ``long` `long` `c1 = 0, c2 = 0; ` ` `  `        ``// if the current element is divisible by k, ` `        ``// count elements in left hash. ` `        ``if` `(a[i] % r == 0) ` `            ``c1 = left[a[i] / r]; ` ` `  `        ``// decrease the count in right hash ` `        ``right[a[i]]--; ` ` `  `        ``// number of right elements  ` `        ``c2 = right[a[i] * r]; ` ` `  `        ``// calculate the answer ` `        ``ans += c1 * c2; ` ` `  `        ``left[a[i]]++; ``// left count of a[i] ` `    ``} ` ` `  `    ``// returns answer ` `    ``return` `ans; ` `} ` ` `  `// driver program  ` `int` `main() ` `{ ` `    ``int` `a[] = { 1, 2, 6, 2, 3, 6, 9, 18, 3, 9 }; ` `    ``int` `n = ``sizeof``(a) / ``sizeof``(a); ` `    ``int` `r = 3; ` `    ``cout << subsequences(a, n, r); ` `    ``return` `0; ` `} `

## Python3

 `# Python3 program to count GP subsequences  ` `# of size 3.  ` `from` `collections ``import` `defaultdict ` ` `  `# Returns count of G.P. subseqeunces  ` `# with length 3 and common ratio r  ` `def` `subsequences(a, n, r):  ` ` `  `    ``# hashing to maintain left and right ` `    ``# array elements to the main count  ` `    ``left ``=` `defaultdict(``lambda``:``0``) ` `    ``right ``=` `defaultdict(``lambda``:``0``) ` ` `  `    ``# stores the answer  ` `    ``ans ``=` `0` ` `  `    ``# traverse through the elements  ` `    ``for` `i ``in` `range``(``0``, n):  ` `        ``right[a[i]] ``+``=` `1` `# keep the count in the hash  ` ` `  `    ``# traverse through all elements and  ` `    ``# find out the number of elements as k1*k2  ` `    ``for` `i ``in` `range``(``0``, n):  ` ` `  `        ``# keep the count of left and right elements  ` `        ``# left is a[i]/r and right a[i]*r  ` `        ``c1, c2 ``=` `0``, ``0` ` `  `        ``# if the current element is divisible  ` `        ``# by k, count elements in left hash.  ` `        ``if` `a[i] ``%` `r ``=``=` `0``:  ` `            ``c1 ``=` `left[a[i] ``/``/` `r]  ` ` `  `        ``# decrease the count in right hash  ` `        ``right[a[i]] ``-``=` `1` ` `  `        ``# number of right elements  ` `        ``c2 ``=` `right[a[i] ``*` `r]  ` ` `  `        ``# calculate the answer  ` `        ``ans ``+``=` `c1 ``*` `c2  ` ` `  `        ``left[a[i]] ``+``=` `1` `# left count of a[i]  ` ` `  `    ``return` `ans  ` ` `  `# Driver Code ` `if` `__name__ ``=``=` `"__main__"``:  ` ` `  `    ``a ``=` `[``1``, ``2``, ``6``, ``2``, ``3``, ``6``, ``9``, ``18``, ``3``, ``9``]  ` `    ``n ``=` `len``(a)  ` `    ``r ``=` `3` `    ``print``(subsequences(a, n, r))  ` ` `  `# This code is contributed by  ` `# Rituraj Jain `

Output:

```6
```

Time complexity: O(n)

My Personal Notes arrow_drop_up Striver(underscore)79 at Codechef and codeforces D

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