TLDR: Here's a constant-time solution: http://codepad.org/Id9eu0Ax

Matching the the sequence of fizz/buzz/fizbuzz disregarding numbers is a repetitive case.

Start with

    3 Fizz
    5 Buzz
    6 Fizz
    9 Fizz
    10 Buzz
    12 Fizz
    15 FizzBuzz

    18 Fizz
    20 Buzz
    21 Fizz
    24 Fizz
    25 Buzz
    27 Fizz
    30 FizzBuzz
    
    [repeats]

Just split the input list as a prefix of a few, and then a repeat. The goal of the shortest total distance between them is constant. You just need to measure the number of ints between each of the 8 repeating distances.

For example:

     3 ->  Fizz
    +2 ->  Buzz
    +1 ->  Fizz
    +3 ->  Fizz
    +1 ->  Buzz
    +2 ->  Fizz
    +3 ->  FizzBuzz

    +3 ->  Fizz
    +2 ->  Buzz
    +1 ->  Fizz
    +3 ->  Fizz
    +1 ->  Buzz
    +2 ->  Fizz
    +3 ->  FizzBuzz

So you would match against the list [Fizz, Buzz, Fizz, Fizz, Buzz, Fizz, FizzBuzz].