Add solution for the euler project problem 137. (#12694)

project_euler/problem_137/sol1.py

@@ -0,0 +1,39 @@
+"""
+Project Euler Problem 137: https://projecteuler.net/problem=137
+
+Fibonacci Golden Nuggets
+
+The polynomial sequence can be rewritten in the finite form:
+
+A_F(x) = x / (1 - x - x^2)
+
+And then the problem is to solve for it for rational x that give A_F(x) as positive
+integer. It turns out that the solution is for the n-th golden nugget is given by
+F(2n) * F(2n + 1), where F(k) is the k'th Fibonacci number.
+
+Reference: https://oeis.org/A081018
+
+"""
+
+
+def solution(n: int = 15) -> int:
+    """
+    It calculates fibonachi numbers 2n and 2n+1, and returns their product.
+
+    >>> solution(3)
+    104
+    >>> solution(10)
+    74049690
+    """
+
+    k = 2 * n
+
+    fib1 = fib2 = 1
+    for _ in range(k - 1):
+        fib1, fib2 = fib2, fib1 + fib2
+
+    return fib1 * fib2
+
+
+if __name__ == "__main__":
+    print(f"{solution() = }")