Science Forums 2

[CraigD]

Consider the integer sequence beginning with these 512 integers: 1,2,2,2,3,3,3,2,3,4,5,3,4,4,4,2,3,4,5,4,6,6,3,3,5, 5,6,4,5,5,5,2,3,4,5,4,6,6,7,4,5,7,7,6,8,4,4,3,5,6, 5,5,8,7,7,4,6,6,7,5,6,6,6,2,3,4,5,4,6,6,7,4,6,7,9, 6,7,8,5,4,6,6,8,7,7,8,7,6,9,9,5,4,6,5,5,3,5,6,6,6, 9,6,7,5,6,9,8,7,8,8,4,4,7,7,8,6,9,8,8,5,7,7,8,6,7, 7,7,2,3,4,5,4,6,6,7,4,6,7,9,6,8,8,9,4,5,7,7,7,9,10 ,6,6,9,8,10,8,5,6,6,4,6,7,9,6,8,9,9,7,9,8,8,8,9,8, 9,6,8,10,9,9,13,6,6,4,7,7,6,5,7,6,6,3,5,6,6,6,9,7, 8,6,9,10,8,6,6,8,8,5,7,7,10,9,8,9,7,7,11,9,10,8,10 ,5,5,4,7,8,7,7,9,9,7,6,6,10,11,8,10,9,9,5,8,8,9,7, 10,9,9,6,8,8,9,7,8,8,8,2,3,4,5,4,6,6,7,4,6,7,9,6,8 ,8,9,4,6,7,9,7,10,10,5,6,8,9,9,8,9,10,6,4,6,6,8,7, 10,8,10,7,7,10,10,10,9,7,8,6,9,10,7,8,11,11,10,8,1 1,6,8,6,6,7,7,4,6,7,9,7,9,10,8,6,9,9,12,9,8,10,9,7 ,10,10,8,8,9,9,9,8,9,10,10,8,11,10,7,6,9,9,9,10,11 ,10,12,9,10,14,7,6,6,7,6,4,8,8,8,7,11,7,7,5,8,8,7, 6,8,7,7,3,5,6,6,6,9,7,8,6,9,10,9,7,10,9,10,6,7,10, 10,10,7,9,7,6,9,7,10,8,8,9,6,5,8,8,10,7,9,11,11,9, 7,9,8,9,11,8,9,7,9,12,9,9,12,11,9,8,12,11,12,5,7,6 ,6,4,7,8,8,8,8,8,8,7,11,10,12,9,10,8,8,6,8,7,11,10 ,11,12,8,8,11,11,10,9,10,10,5,5,9,9,10,8,11,10,11, 7,8,11,12,9,11,10,10,6,9,9,10,8,11,10,10,7,9,9,10, 8,9,9,9,2

The puzzle is simple: find an expression that generates the sequence.

There’s only one rule: the answer can’t be a degree 511 polynomial that generates these 512 values. You’re welcome to post this answer just to show off your (or your computer’s ) polynomial fitting skills, but it won’t count as a win.

Hints:

• The sequence isn’t (to the best of my knowledge) generatable by a simple polynomial
• I generated it using
  • A generalization of the coefficients of the expected values in this post
  • Something involving base 2 (binary) numbers
• If you know the answer, you’d almost certainly know how to generate the sequence that begins with the following 512 numbers:
  1,1,2,1,2,2,3,1,2,2,3,2,3,3,4,1,2,2,3,2,3,3,4,2,3, 3,4,3,4,4,5,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3, 4,3,4,4,5,3,4,4,5,4,5,5,6,1,2,2,3,2,3,3,4,2,3,3,4, 3,4,4,5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,2,3,3,4,3, 4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6, 6,7,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4, 5,3,4,4,5,4,5,5,6,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6, 3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,2,3,3,4,3,4,4,5,3, 4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4, 4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6, 7,6,7,7,8,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4, 3,4,4,5,3,4,4,5,4,5,5,6,2,3,3,4,3,4,4,5,3,4,4,5,4, 5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,2,3,3,4,3,4, 4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6, 7,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7, 5,6,6,7,6,7,7,8,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3, 4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,4,5,5,6,4,5, 5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,3,4,4, 5,4,5,5,6,4,5,5,6,5,6,6,7,4,5,5,6,5,6,6,7,5,6,6,7, 6,7,7,8,4,5,5,6,5,6,6,7,5,6,6,7,6,7,7,8,5,6,6,7,6, 7,7,8,6,7,7,8,7,8,8,9,1