45 really easy once you can quickly verify pentagonal numbers

45/main.py

@@ -0,0 +1,35 @@
+'''
+Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:
+
+Triangle	 	Tn=n(n+1)/2	 	1, 3, 6, 10, 15, ...
+Pentagonal	 	Pn=n(3n−1)/2	1, 5, 12, 22, 35, ...
+Hexagonal	 	Hn=n(2n−1)	 	1, 6, 15, 28, 45, ...
+It can be verified that T285 = P165 = H143 = 40755.
+
+Find the next triangle number that is also pentagonal and hexagonal.
+'''
+
+# Observe that every other triangle number is also a hexagonal number, so the 1st, the 3rd, 5th, etc.
+# So we only need to check all the uneven n triangle numbers to be pentagonal
+
+# In addition, we can check if a number is pentagonal by checking if sqrt(1 + 24 n) % 6 == 5
+
+import math
+
+def t(n):
+    return n * (n+1) / 2
+
+def isPentagonal(x):
+    return math.sqrt(1 + 24 * x) % 6 == 5
+
+def main():
+    print("Hello this is Patrick")
+
+    n = 287
+    while not isPentagonal(t(n)):
+        n += 2
+    
+    print(t(n))
+
+if __name__ == "__main__":
+    main()