I can totally relate to this. I have to sleep with a fan or some other true white noise in the background. HAVE to. Absolutely have to.
My (ex-)wife bought or was given this noise generator thingy because we had problems sleeping with snoring and TV and such, and when I say "we" I mean that I had a problem sleeping, and she had a problem not turning off the TV when it was time to go to sleep. Anyway, it had a bunch of audio modes to pick from: birds, happy burbling river noise, jungle, crickets, a few others, and a white noise track. Well, the track for each was, as you said, just long enough that I could pick out the patterns, and as we listened to them to try to pick one, I was saying "no" to each one rather quickly, and she was getting more and more frustrated. Finally I said, "just try the white noise one, I don't care if it's boring, it's the only one that's going to work." Rather soon (almost immediately) I realized the track for the white noise was simply an 8 second loop and the "whiteness" JUST barely didn't match up from the end and the beginning so I could hear this tiny clip in the audio when it looped. "NOPE THIS ONE WON'T WORK EITHER" in fact, it was by far the MOST annoying one of the bunch. She never understood why, and it got returned or thrown out or donated or whatever.
But yeah.
Patterns.
They suck sometimes.