AoPS: Weird Equations 2

Let weird(n) be the number of ways to write n as a + a + b + c, where n, a, b, and c are distinct positive integers. Is weird(n) an increasing function?

Surprisingly, weird(n) is not increasing... it sure seems like it should be! See if you can find a pattern for when weird(n) goes down. Can you explain what is going on?

Hint: the author wrote a script to print the first few values of n where weird(n) goes down and a few other things before the pattern became apparent! This problem is a good example of how computers can help us find patterns!