Phi Functions in Go: Calculating the nth Phi Number

The golden ratio, often denoted by the Greek letter phi (φ), is an irrational number approximately equal to 1.6180339887. Phi numbers form a sequence where each number is the sum of the two preceding numbers, similar to the Fibonacci sequence, but with different starting values. This challenge asks you to implement functions in Go to calculate the nth Phi number efficiently.

Problem Description

You are tasked with creating a Go package that provides functions for calculating Phi numbers. Phi numbers are defined by the recurrence relation: φ(n) = φ(n-1) + φ(n-2), with φ(0) = 0 and φ(1) = 1. Your package should include two functions:

Phi(n int) int: This function should calculate the nth Phi number. It should return the calculated value as an integer.
PhiRecursive(n int) int: This function should calculate the nth Phi number using a recursive approach. It should also return the calculated value as an integer. This is primarily for comparison and understanding of different approaches.

The functions should handle edge cases gracefully, such as negative input values. For negative input, both functions should return 0.

Examples

Example 1:

Input: n = 0
Output: 0
Explanation: φ(0) is defined as 0.

Example 2:

Input: n = 1
Output: 1
Explanation: φ(1) is defined as 1.

Example 3:

Input: n = 5
Output: 8
Explanation: φ(2) = 1, φ(3) = 2, φ(4) = 3, φ(5) = 5, φ(6) = 8.

Example 4:

Input: n = -2
Output: 0
Explanation: Negative input should return 0.

Constraints

n will be an integer.
The result of Phi(n) and PhiRecursive(n) should be an integer.
For n >= 0, the functions should return the correct Phi number.
For n < 0, the functions should return 0.
The Phi(n) function should be optimized for performance (avoiding unnecessary recursion). While PhiRecursive(n) is allowed to be recursive, Phi(n) should use an iterative approach.
The maximum value of n for which you need to calculate the Phi number is 40. (This is to prevent integer overflow issues with larger values).

Notes

Consider using an iterative approach for the Phi(n) function to improve performance. Recursion can be inefficient for larger values of n.
The PhiRecursive(n) function is primarily for demonstration and comparison. It doesn't need to be highly optimized.
Think about how to handle potential integer overflow issues, although the constraint on n limits the scope of this concern.
Your solution should be well-documented and easy to understand. Include comments to explain your code.
Create a Go package named phi to contain your functions. The package should be importable.

Phi Functions in Go: Calculating the nth Phi Number

Problem Description

Phi(n int) int: This function should calculate the nth Phi number. It should return the calculated value as an integer.

PhiRecursive(n int) int: This function should calculate the nth Phi number using a recursive approach. It should also return the calculated value as an integer. This is primarily for comparison and understanding of different approaches.

The functions should handle edge cases gracefully, such as negative input values. For negative input, both functions should return 0.

Examples

Example 1:

Input: n = 0 Output: 0 Explanation: φ(0) is defined as 0.

Example 2:

Input: n = 1 Output: 1 Explanation: φ(1) is defined as 1.

Example 3:

Input: n = 5 Output: 8 Explanation: φ(2) = 1, φ(3) = 2, φ(4) = 3, φ(5) = 5, φ(6) = 8.

Example 4:

Input: n = -2 Output: 0 Explanation: Negative input should return 0.

Constraints

n will be an integer.

The result of Phi(n) and PhiRecursive(n) should be an integer.

For n >= 0, the functions should return the correct Phi number.

For n < 0, the functions should return 0.

The Phi(n) function should be optimized for performance (avoiding unnecessary recursion). While PhiRecursive(n) is allowed to be recursive, Phi(n) should use an iterative approach.

The maximum value of n for which you need to calculate the Phi number is 40. (This is to prevent integer overflow issues with larger values).

Notes

Consider using an iterative approach for the Phi(n) function to improve performance. Recursion can be inefficient for larger values of n.

The PhiRecursive(n) function is primarily for demonstration and comparison. It doesn't need to be highly optimized.

Think about how to handle potential integer overflow issues, although the constraint on n limits the scope of this concern.

Your solution should be well-documented and easy to understand. Include comments to explain your code.

Create a Go package named phi to contain your functions. The package should be importable.