Given a number K, the task is to find the sum of numbers at the Kth level of the Fibonacci triangle.
Examples:
Input: K = 3 Output: 10 Explanation: Fibonacci triangle till level 3: 0 1 1 2 3 5 Sum at 3rd level = 2 + 3 + 5 = 10 Input: K = 2 Output: 2 Explanation: Fibonacci triangle till level 3: 0 1 1 Sum at 3rd level = 1 + 1 = 2
Approach:
- Till Kth level, i.e. from level [1, K-1], count of Fibonacci numbers already used can be computed as:
cnt = N(Level 1) + N(Level 2) + N(Level 3) + ... + N(Level K-1) = 1 + 2 + 3 + ... + (K-1) = K*(K-1)/2
- Also, we know that the Kth level will contain K Fibonacci numbers.
- Therefore we can find the numbers in the Kth level as Fibonacci numbers in the range [(cnt + 1), (cnt + 1 + K)].
- We can find the sum of Fibonacci numbers in a range in O(1) time using Binet’s Formula.
Below is the implementation of the above approach:
C++
// C++ implementation to find // the Sum of numbers in the // Kth level of a Fibonacci triangle #include <bits/stdc++.h> using namespace std;
#define MAX 1000000 // Function to return // the nth Fibonacci number int fib( int n)
{ double phi = (1 + sqrt (5)) / 2;
return round( pow (phi, n) / sqrt (5));
} // Function to return // the required sum of the array int calculateSum( int l, int r)
{ // Using our deduced result
int sum = fib(r + 2) - fib(l + 1);
return sum;
} // Function to return the sum of // fibonacci in the Kth array int sumFibonacci( int k)
{ // Count of fibonacci which are in
// the arrays from 1 to k - 1
int l = (k * (k - 1)) / 2;
int r = l + k;
int sum = calculateSum(l, r - 1);
return sum;
} // Driver code int main()
{ int k = 3;
cout << sumFibonacci(k);
return 0;
} |
Java
// Java implementation to find // the Sum of numbers in the // Kth level of a Fibonacci triangle import java.util.*;
class GFG
{ // Function to return // the nth Fibonacci number static int fib( int n)
{ double phi = ( 1 + Math.sqrt( 5 )) / 2 ;
return ( int )Math.round(Math.pow(phi, n) / Math.sqrt( 5 ));
} // Function to return // the required sum of the array static int calculateSum( int l, int r)
{ // Using our deduced result
int sum = fib(r + 2 ) - fib(l + 1 );
return sum;
} // Function to return the sum of // fibonacci in the Kth array static int sumFibonacci( int k)
{ // Count of fibonacci which are in
// the arrays from 1 to k - 1
int l = (k * (k - 1 )) / 2 ;
int r = l + k;
int sum = calculateSum(l, r - 1 );
return sum;
} // Driver code public static void main(String args[])
{ int k = 3 ;
System.out.println(sumFibonacci(k));
} } // This code is contributed by AbhiThakur |
Python3
# Python3 implementation to find # the Sum of numbers in the # Kth level of a Fibonacci triangle import math
MAX = 1000000
# Function to return # the nth Fibonacci number def fib(n):
phi = ( 1 + math.sqrt( 5 )) / 2
return round ( pow (phi, n) / math.sqrt( 5 ))
# Function to return # the required sum of the array def calculateSum(l, r):
# Using our deduced result
sum = fib(r + 2 ) - fib(l + 1 )
return sum
# Function to return the sum of # fibonacci in the Kth array def sumFibonacci(k) :
# Count of fibonacci which are in
# the arrays from 1 to k - 1
l = (k * (k - 1 )) / 2
r = l + k
sum = calculateSum(l, r - 1 )
return sum
# Driver code k = 3
print (sumFibonacci(k))
# This code is contributed by Sanjit_Prasad |
C#
// C# implementation to find // the Sum of numbers in the // Kth level of a Fibonacci triangle using System;
class GFG
{ // Function to return // the nth Fibonacci number static int fib( int n)
{ double phi = (1 + Math.Sqrt(5)) / 2;
return ( int )Math.Round(Math.Pow(phi, n) / Math.Sqrt(5));
} // Function to return // the required sum of the array static int calculateSum( int l, int r)
{ // Using our deduced result
int sum = fib(r + 2) - fib(l + 1);
return sum;
} // Function to return the sum of // fibonacci in the Kth array static int sumFibonacci( int k)
{ // Count of fibonacci which are in
// the arrays from 1 to k - 1
int l = (k * (k - 1)) / 2;
int r = l + k;
int sum = calculateSum(l, r - 1);
return sum;
} // Driver code public static void Main()
{ int k = 3;
Console.Write(sumFibonacci(k));
} } // This code is contributed by mohit kumar 29 |
Javascript
<script> // Javascript implementation to find // the Sum of numbers in the // Kth level of a Fibonacci triangle // Function to return
// the nth Fibonacci number
function fib(n) {
var phi = (1 + Math.sqrt(5)) / 2;
return parseInt( Math.round
(Math.pow(phi, n) / Math.sqrt(5)));
}
// Function to return
// the required sum of the array
function calculateSum(l , r) {
// Using our deduced result
var sum = fib(r + 2) - fib(l + 1);
return sum;
}
// Function to return the sum of
// fibonacci in the Kth array
function sumFibonacci(k)
{
// Count of fibonacci which are in
// the arrays from 1 to k - 1
var l = (k * (k - 1)) / 2;
var r = l + k;
var sum = calculateSum(l, r - 1);
return sum;
}
// Driver code
var k = 3;
document.write(sumFibonacci(k));
// This code is contributed by todaysgaurav </script> |
Output:
10
Time Complexity: O((log n) + (n1/2))
Auxiliary Space: O(1)