I was having difficulty seeing Nathaniel Shar's (interesting!) pattern. This might be easier to interpret:
      fib500 inverted http://cs.smith.edu/%7Eorourke/MathOverflow/fib500.gif
(I thought it too intrusive to add this directly to his answer.)

I was having difficulty seeing Nathaniel Shar's (interesting!) pattern. This might be easier to interpret:
     
(I thought it too intrusive to add this directly to his answer.)