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Number of quadruples where the first three terms are in AP and last three terms are in GP
• Last Updated : 31 Jan, 2020

Given an array arr[] of N integers. The task is to find the number of index quadruples (i, j, k, l) such that a[i], a[j] and a[k] are in AP and a[j], a[k] and a[l] are in GP. All the quadruples have to be distinct.

Examples:

Input: arr[] = {2, 6, 4, 9, 2}
Output: 2
Indexes of elements in the quadruples are (0, 2, 1, 3) and (4, 2, 1, 3) and corresponding quadruples are (2, 4, 6, 9) and (2, 4, 6, 9)

Input: arr[] = {1, 1, 1, 1}
Output: 24

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

A naive approach is to solve the above problem using four nested loops. Check for the first three elements if they are in AP or not and then check whether the last three elements are in GP or not. If both the conditions satisfy, then they increase the count by 1.

Time Complexity: O(n4)

An efficient approach is to use combinatorics to solve the above problem. Initially keep a count of the number of occurrences of every array element. Run two nested loops, and consider both elements to be the second and third number. Hence the first element will be a[j] – (a[k] – a[j]) and the fourth element will be a[k] * a[k] / a[j] if it is an integer value. Hence the number of quadruples using this two index j and k will be count of first number * count of fourth number with the second and third element being fixed.

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach``#include ``using` `namespace` `std;`` ` `// Function to return the count of quadruples``int` `countQuadruples(``int` `a[], ``int` `n)``{`` ` `    ``// Hash table to count the number of occurrences``    ``unordered_map<``int``, ``int``> mpp;`` ` `    ``// Traverse and increment the count``    ``for` `(``int` `i = 0; i < n; i++)``        ``mpp[a[i]]++;`` ` `    ``int` `count = 0;`` ` `    ``// Run two nested loop for second and third element``    ``for` `(``int` `j = 0; j < n; j++) {``        ``for` `(``int` `k = 0; k < n; k++) {`` ` `            ``// If they are same``            ``if` `(j == k)``                ``continue``;`` ` `            ``// Initially decrease the count``            ``mpp[a[j]]--;``            ``mpp[a[k]]--;`` ` `            ``// Find the first element using common difference``            ``int` `first = a[j] - (a[k] - a[j]);`` ` `            ``// Find the fourth element using GP``            ``// y^2 = x * z property``            ``int` `fourth = (a[k] * a[k]) / a[j];`` ` `            ``// If it is an integer``            ``if` `((a[k] * a[k]) % a[j] == 0) {`` ` `                ``// If not equal``                ``if` `(a[j] != a[k])``                    ``count += mpp[first] * mpp[fourth];`` ` `                ``// Same elements``                ``else``                    ``count += mpp[first] * (mpp[fourth] - 1);``            ``}`` ` `            ``// Later increase the value for``            ``// future calculations``            ``mpp[a[j]]++;``            ``mpp[a[k]]++;``        ``}``    ``}``    ``return` `count;``}`` ` `// Driver code``int` `main()``{``    ``int` `a[] = { 2, 6, 4, 9, 2 };``    ``int` `n = ``sizeof``(a) / ``sizeof``(a);`` ` `    ``cout << countQuadruples(a, n);`` ` `    ``return` `0;``}`

## Java

 `// Java implementation of the approach``import` `java.util.*;`` ` `class` `GFG``{`` ` `    ``// Function to return the count of quadruples``    ``static` `int` `countQuadruples(``int` `a[], ``int` `n) ``    ``{`` ` `        ``// Hash table to count the number of occurrences``        ``HashMap mp = ``new` `HashMap();`` ` `        ``// Traverse and increment the count``        ``for` `(``int` `i = ``0``; i < n; i++)``            ``if` `(mp.containsKey(a[i]))``            ``{``                ``mp.put(a[i], mp.get(a[i]) + ``1``);``            ``}``            ``else``            ``{``                ``mp.put(a[i], ``1``);``            ``}`` ` `        ``int` `count = ``0``;`` ` `        ``// Run two nested loop for second and third element``        ``for` `(``int` `j = ``0``; j < n; j++)``        ``{``            ``for` `(``int` `k = ``0``; k < n; k++)``            ``{`` ` `                ``// If they are same``                ``if` `(j == k)``                    ``continue``;`` ` `                ``// Initially decrease the count``                ``mp.put(a[j], mp.get(a[j]) - ``1``);``                ``mp.put(a[k], mp.get(a[k]) - ``1``);`` ` `                ``// Find the first element using common difference``                ``int` `first = a[j] - (a[k] - a[j]);`` ` `                ``// Find the fourth element using GP``                ``// y^2 = x * z property``                ``int` `fourth = (a[k] * a[k]) / a[j];`` ` `                ``// If it is an integer``                ``if` `((a[k] * a[k]) % a[j] == ``0``)``                ``{`` ` `                    ``// If not equal``                    ``if` `(a[j] != a[k]) ``                    ``{``                        ``if` `(mp.containsKey(first) && mp.containsKey(fourth))``                            ``count += mp.get(first) * mp.get(fourth);``                    ``}``                     ` `                    ``// Same elements``                    ``else` `if` `(mp.containsKey(first) && mp.containsKey(fourth))``                        ``count += mp.get(first) * (mp.get(fourth) - ``1``);``                ``}`` ` `                ``// Later increase the value for``                ``// future calculations``                ``if` `(mp.containsKey(a[j]))``                ``{``                    ``mp.put(a[j], mp.get(a[j]) + ``1``);``                ``} ``                ``else``                ``{``                    ``mp.put(a[j], ``1``);``                ``}``                ``if` `(mp.containsKey(a[k]))``                ``{``                    ``mp.put(a[k], mp.get(a[k]) + ``1``);``                ``} ``                ``else` `                ``{``                    ``mp.put(a[k], ``1``);``                ``}``            ``}``        ``}``        ``return` `count;``    ``}`` ` `    ``// Driver code``    ``public` `static` `void` `main(String[] args)``    ``{``        ``int` `a[] = { ``2``, ``6``, ``4``, ``9``, ``2` `};``        ``int` `n = a.length;`` ` `        ``System.out.print(countQuadruples(a, n));``    ``}``}`` ` `// This code is contributed by 29AjayKumar`

## Python3

 `# Python3 implementation of the approach `` ` `# Function to return the count of quadruples ``def` `countQuadruples(a, n) : `` ` `    ``# Hash table to count the number ``    ``# of occurrences ``    ``mpp ``=` `dict``.fromkeys(a, ``0``); `` ` `    ``# Traverse and increment the count ``    ``for` `i ``in` `range``(n) :``        ``mpp[a[i]] ``+``=` `1``; `` ` `    ``count ``=` `0``; `` ` `    ``# Run two nested loop for second``    ``# and third element ``    ``for` `j ``in` `range``(n) : ``        ``for` `k ``in` `range``(n) : `` ` `            ``# If they are same ``            ``if` `(j ``=``=` `k) :``                ``continue``; `` ` `            ``# Initially decrease the count ``            ``mpp[a[j]] ``-``=` `1``; ``            ``mpp[a[k]] ``-``=` `1``; `` ` `            ``# Find the first element using``            ``# common difference ``            ``first ``=` `a[j] ``-` `(a[k] ``-` `a[j]);``             ` `            ``if` `first ``not` `in` `mpp :``                ``mpp[first] ``=` `0``;``                 ` `            ``# Find the fourth element using ``            ``# GP y^2 = x * z property ``            ``fourth ``=` `(a[k] ``*` `a[k]) ``/``/` `a[j];``             ` `            ``if` `fourth ``not` `in` `mpp :``                ``mpp[fourth] ``=` `0``;``                 ` `            ``# If it is an integer ``            ``if` `((a[k] ``*` `a[k]) ``%` `a[j] ``=``=` `0``) :`` ` `                ``# If not equal ``                ``if` `(a[j] !``=` `a[k]) :``                    ``count ``+``=` `mpp[first] ``*` `mpp[fourth]; `` ` `                ``# Same elements ``                ``else` `:``                    ``count ``+``=` `(mpp[first] ``*` `                             ``(mpp[fourth] ``-` `1``)); ``             ` `            ``# Later increase the value for ``            ``# future calculations ``            ``mpp[a[j]] ``+``=` `1``; ``            ``mpp[a[k]] ``+``=` `1``;``             ` `    ``return` `count; `` ` `# Driver code ``if` `__name__ ``=``=` `"__main__"` `:`` ` `    ``a ``=` `[ ``2``, ``6``, ``4``, ``9``, ``2` `]; ``    ``n ``=` `len``(a) ; `` ` `    ``print``(countQuadruples(a, n)); `` ` `# This code is contributed by Ryuga`

## C#

 `// C# implementation of the approach``using` `System;``using` `System.Collections.Generic;`` ` `class` `GFG``{`` ` `    ``// Function to return the count of quadruples``    ``static` `int` `countQuadruples(``int` `[]a, ``int` `n) ``    ``{`` ` `        ``// Hash table to count the number of occurrences``        ``Dictionary<``int``, ``int``> mp = ``new` `Dictionary<``int``, ``int``>();`` ` `        ``// Traverse and increment the count``        ``for` `(``int` `i = 0; i < n; i++)``            ``if` `(mp.ContainsKey(a[i]))``            ``{``                ``mp[a[i]] = mp[a[i]] + 1;``            ``}``            ``else``            ``{``                ``mp.Add(a[i], 1);``            ``}`` ` `        ``int` `count = 0;`` ` `        ``// Run two nested loop for second and third element``        ``for` `(``int` `j = 0; j < n; j++)``        ``{``            ``for` `(``int` `k = 0; k < n; k++)``            ``{`` ` `                ``// If they are same``                ``if` `(j == k)``                    ``continue``;`` ` `                ``// Initially decrease the count``                ``mp[a[j]] = mp[a[j]] - 1;``                ``mp[a[k]] = mp[a[k]] - 1;`` ` `                ``// Find the first element using common difference``                ``int` `first = a[j] - (a[k] - a[j]);`` ` `                ``// Find the fourth element using GP``                ``// y^2 = x * z property``                ``int` `fourth = (a[k] * a[k]) / a[j];`` ` `                ``// If it is an integer``                ``if` `((a[k] * a[k]) % a[j] == 0)``                ``{`` ` `                    ``// If not equal``                    ``if` `(a[j] != a[k]) ``                    ``{``                        ``if` `(mp.ContainsKey(first) && mp.ContainsKey(fourth))``                            ``count += mp[first] * mp[fourth];``                    ``}``                     ` `                    ``// Same elements``                    ``else` `if` `(mp.ContainsKey(first) && mp.ContainsKey(fourth))``                        ``count += mp[first] * (mp[fourth] - 1);``                ``}`` ` `                ``// Later increase the value for``                ``// future calculations``                ``if` `(mp.ContainsKey(a[j]))``                ``{``                    ``mp[a[j]] = mp[a[j]] + 1;``                ``} ``                ``else``                ``{``                    ``mp.Add(a[j], 1);``                ``}``                ``if` `(mp.ContainsKey(a[k]))``                ``{``                    ``mp[a[k]] = mp[a[k]] + 1;``                ``} ``                ``else``                ``{``                    ``mp.Add(a[k], 1);``                ``}``            ``}``        ``}``        ``return` `count;``    ``}`` ` `    ``// Driver code``    ``public` `static` `void` `Main(String[] args)``    ``{``        ``int` `[]a = { 2, 6, 4, 9, 2 };``        ``int` `n = a.Length;`` ` `        ``Console.Write(countQuadruples(a, n));``    ``}``}`` ` `// This code is contributed by 29AjayKumar`
Output:
```2
```

Time Complexity: O(N2)
Auxiliary Space: O(N)

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