mirror of
https://github.com/TheAlgorithms/Ruby
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45 lines
1.2 KiB
Ruby
45 lines
1.2 KiB
Ruby
# The Fibonacci numbers, commonly denoted F(n) form a sequence,
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# called the Fibonacci sequence, such that each number is the sum
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# of the two preceding ones, starting from 0 and 1. That is,
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#
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# F(0) = 0, F(1) = 1
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# F(n) = F(n - 1) + F(n - 2), for n > 1.
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#
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# Given n, calculate F(n).
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#
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# Approach: Math
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#
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# Intuition: Using the golden ratio, a.k.a Binet's formula
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# Algorithm: Use the golden ratio formula to calculate the Nth Fibonacci number.
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# https://demonstrations.wolfram.com/GeneralizedFibonacciSequenceAndTheGoldenRatio/
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# Complexity Analysis
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# Time complexity: O(1). Constant time complexity since we are using no loops or recursion
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# and the time is based on the result of performing the calculation using Binet's formula.
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#
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# Space complexity: O(1). The space used is the space needed to create the variable
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# to store the golden ratio formula.
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def fibonacci(n)
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golden_ratio = (1 + 5**0.5) / 2
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((golden_ratio**n + 1) / 5**0.5).to_i
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end
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n = 2
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puts(fibonacci(n))
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# Output: 1
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# Explanation: F(2) = F(1) + F(0) = 1 + 0 = 1.
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n = 3
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puts(fibonacci(n))
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# Output: 2
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# Explanation: F(3) = F(2) + F(1) = 1 + 1 = 2.
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n = 4
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puts(fibonacci(n))
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# Output: 3
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# Explanation: F(4) = F(3) + F(2) = 2 + 1 = 3.
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