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Sum of all Non-Fibonacci numbers in a range for Q queries
• Difficulty Level : Medium
• Last Updated : 12 Apr, 2021

Given Q queries containing ranges in the form of [L, R], the task is to find the sum of all non-fibonacci numbers for each range in the given queries.
Examples:

Input: arr[][] = {{1, 5}, {6, 10}}
Output: 4 32
Explanation:
Query 1: In the range [1, 5], only 4 is a non-fibonacci number.
Query 2: In the range [6, 10], 6, 7, 9 and 10 are the non-fibonacci numbers.
Therefore, 6 + 7 + 9 + 10 = 32.
Input: arr[][] = {{10, 20}, {20, 50}}
Output: 152 10792

Approach: The idea is to use a prefix sum array. The sum of all non-fibonacci numbers is precomputed and stored in an array. So that every query can be answered in O(1) time. Every index of the array stores the sum of all non-fibonacci numbers from 1 to that index. So for finding the sum of all non-fibonacci numbers in a range can be computed as:

```Let the precomputed array is stored in pref[] array
sum = pref[R] - pref[L - 1]```

Below is the implementation of the above approach:

## C++

 `// C++ implementation to find the``// sum of all non-fibonacci numbers``// in a range from L to R` `#include ``#define ll int``using` `namespace` `std;` `// Array to precompute the sum of``// non-fibonacci numbers``long` `long` `pref[100010];` `// Function to find if a number``// is a perfect square``bool` `isPerfectSquare(``int` `x)``{``    ``int` `s = ``sqrt``(x);``    ``return` `(s * s == x);``}` `// Function that returns N``// if N is non-fibonacci number``int` `isNonFibonacci(``int` `n)``{``    ``// N is Fibinacci if one of``    ``// 5*n*n + 4 or 5*n*n - 4 or both``    ``// are perferct square``    ``if` `(isPerfectSquare(5 * n * n + 4)``        ``|| isPerfectSquare(5 * n * n - 4))``        ``return` `0;``    ``else``        ``return` `n;``}` `// Function to precompute sum of``// non-fibonacci Numbers``void` `compute()``{``    ``for` `(``int` `i = 1; i <= 100000; ++i) {``        ``pref[i] = pref[i - 1]``                  ``+ isNonFibonacci(i);``    ``}``}` `// Function to find the sum of all``// non-fibonacci numbers in a range``void` `printSum(``int` `L, ``int` `R)``{``    ``int` `sum = pref[R] - pref[L - 1];``    ``cout << sum << ``" "``;``}` `// Driver Code``int` `main()``{``    ``// Pre-computation``    ``compute();` `    ``int` `Q = 2;``    ``int` `arr[][2] = { { 1, 5 },``                     ``{ 6, 10 } };``    ``// Loop to find the sum for``    ``// each query``    ``for` `(``int` `i = 0; i < Q; i++) {``        ``printSum(arr[i][0], arr[i][1]);``    ``}``    ``return` `0;``}`

## Java

 `// Java implementation to find the``// sum of all non-fibonacci numbers``// in a range from L to R``import` `java.util.*;`` ` `// Array to precompute the sum of``// non-fibonacci numbers` `class` `GFG``{``static` `long` `pref[] = ``new` `long``[``100010``];`` ` `// Function to find if a number``// is a perfect square``static` `boolean` `isPerfectSquare(``int` `x)``{``    ``int` `s =(``int``)Math.sqrt(x);``    ``return` `(s * s == x);``}`` ` `// Function that returns N``// if N is non-fibonacci number``static` `int` `isNonFibonacci(``int` `n)``{``    ``// N is Fibinacci if one of``    ``// 5*n*n + 4 or 5*n*n - 4 or both``    ``// are perferct square``    ``if` `(isPerfectSquare(``5` `* n * n + ``4``)``        ``|| isPerfectSquare(``5` `* n * n - ``4``))``        ``return` `0``;``    ``else``        ``return` `n;``}`` ` `// Function to precompute sum of``// non-fibonacci Numbers``static` `void` `compute()``{``    ``for` `(``int` `i = ``1``; i <= ``100000``; ++i) {``        ``pref[i] = pref[i - ``1``]``                  ``+ isNonFibonacci(i);``    ``}``}`` ` `// Function to find the sum of all``// non-fibonacci numbers in a range``static` `void` `printSum(``int` `L, ``int` `R)``{``    ``int` `sum = (``int``)(pref[R] - pref[L - ``1``]);``    ``System.out.print(sum + ``" "``);``}`` ` `// Driver Code``public` `static` `void` `main(String []args)``{``    ``// Pre-computation``    ``compute();`` ` `    ``int` `Q = ``2``;``    ``int` `arr[][] = { { ``1``, ``5` `},``                     ``{ ``6``, ``10` `} };``    ``// Loop to find the sum for``    ``// each query``    ``for` `(``int` `i = ``0``; i < Q; i++) {``        ``printSum(arr[i][``0``], arr[i][``1``]);``    ``}``}``}` `// This code is contributed by chitranayal`

## Python3

 `# Python3 implementation to find the``# sum of all non-fibonacci numbers``# in a range from L to R``from` `math ``import` `sqrt` `# Array to precompute the sum of``# non-fibonacci numbers``pref ``=` `[``0``]``*``100010` `# Function to find if a number``# is a perfect square``def` `isPerfectSquare(x):``    ` `    ``s ``=` `int``(sqrt(x))``    ``if` `(s ``*` `s ``=``=` `x):``        ``return` `True``    ``return` `False` `# Function that returns N``# if N is non-fibonacci number``def` `isNonFibonacci(n):``    ` `    ``# N is Fibinacci if one of``    ``# 5*n*n + 4 or 5*n*n - 4 or both``    ``# are perferct square``    ``x ``=` `5` `*` `n ``*` `n``    ``if` `(isPerfectSquare(x ``+` `4``) ``or` `isPerfectSquare(x ``-` `4``)):``        ``return` `0``    ``else``:``        ``return` `n` `# Function to precompute sum of``# non-fibonacci Numbers``def` `compute():``    ` `    ``for` `i ``in` `range``(``1``,``100001``):``        ``pref[i] ``=` `pref[i ``-` `1``] ``+` `isNonFibonacci(i)``    ` `# Function to find the sum of all``# non-fibonacci numbers in a range``def` `printSum(L, R):``    ` `    ``sum` `=` `pref[R] ``-` `pref[L``-``1``]``    ``print``(``sum``, end``=``" "``)` `# Driver Code``# Pre-computation``compute()` `Q ``=` `2``arr ``=` `[[``1``, ``5``],[``6``, ``10``]]``# Loop to find the sum for``# each query` `for` `i ``in` `range``(Q):``    ``printSum(arr[i][``0``], arr[i][``1``])` `# This code is contributed by shubhamsingh10`

## C#

 `// C# implementation to find the``// sum of all non-fibonacci numbers``// in a range from L to R``using` `System;`` ` `// Array to precompute the sum of``// non-fibonacci numbers``class` `GFG``{``static` `long` `[]pref = ``new` `long``[100010];``  ` `// Function to find if a number``// is a perfect square``static` `bool` `isPerfectSquare(``int` `x)``{``    ``int` `s =(``int``)Math.Sqrt(x);``    ``return` `(s * s == x);``}``  ` `// Function that returns N``// if N is non-fibonacci number``static` `int` `isNonFibonacci(``int` `n)``{``    ``// N is Fibinacci if one of``    ``// 5*n*n + 4 or 5*n*n - 4 or both``    ``// are perferct square``    ``if` `(isPerfectSquare(5 * n * n + 4)``        ``|| isPerfectSquare(5 * n * n - 4))``        ``return` `0;``    ``else``        ``return` `n;``}``  ` `// Function to precompute sum of``// non-fibonacci Numbers``static` `void` `compute()``{``    ``for` `(``int` `i = 1; i <= 100000; ++i) {``        ``pref[i] = pref[i - 1]``                  ``+ isNonFibonacci(i);``    ``}``}``  ` `// Function to find the sum of all``// non-fibonacci numbers in a range``static` `void` `printSum(``int` `L, ``int` `R)``{``    ``int` `sum = (``int``)(pref[R] - pref[L - 1]);``    ``Console.Write(sum + ``" "``);``}``  ` `// Driver Code``public` `static` `void` `Main(String []args)``{``    ``// Pre-computation``    ``compute();``  ` `    ``int` `Q = 2;``    ``int` `[,]arr = { { 1, 5 },``                     ``{ 6, 10 } };``    ``// Loop to find the sum for``    ``// each query``    ``for` `(``int` `i = 0; i < Q; i++) {``        ``printSum(arr[i,0], arr[i,1]);``    ``}``}``}` `// This code is contributed by Rajput-Ji`

## Javascript

 ``
Output:
`4 32`

Performance Analysis:

• Time Complexity: As in the above approach, There is pre-computation which takes O(N) time and to answer each query it takes O(1) time.
• Auxiliary Space Complexity: As in the above approach, There is extra space used to precompute the sum of all non-fibonacci number. Hence the auxiliary space complexity will be O(N).

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