Puzzle Sequences

Puzzle Sequences From

The On-Line Encyclopedia of Integer Sequences

Can you guess the rules for generating these sequences?
Note that I believe in giving at least ten terms as a hint.

• This one was in the New York Times a while back:
  2, 3, 3, 5, 10, 13, 39, 43, 172, 177, ... Give up?
• 0, 0, 0, 0, 4, 9, 5, 1, 1, 0, 55, ...
  Hint: write out
  one, two, three, four, five, six, seven, eight, nine, ten, eleven, ...
  and think like a Roman! Give up?
• From Chess Life, said to be a sequence that the world chess champion did not guess (except I bet he was not given 10 terms):
  7, 9, 40, 74, 1526, 5436, 2323240, 29548570, 5397414549030, 873117986721660, ... Give up?
• Everyone knows about the even numbers, sequence A005843.
  Less well-known are the eban numbers (the name is a strong hint!):
  2, 4, 6, 30, 32, 34, 36, 40, 42, 44, 46, ... Give up?
• Somewhat in the same spirit are the emirps:
  13, 17, 31, 37, 71, 73, 79, 97, 107, 113, ... Give up?
• Elegant, classic:
  1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, ... Give up?
• What about this one?
  1, 3, 7, 12, 18, 26, 35, 45, 56, 69, 83, ...
  Hmmm! Still thinking? I've seen nine-year-olds guess it quicker than that. Give up?
• This one is easy for smart seventeen-year-olds, or if you've seen it before, otherwise not!
  2, 12, 1112, 3112, 132112, 1113122112, 311311222112, 13211321322112, 1113122113121113222112, 31131122211311123113322112, ... Give up?
  John Conway's astonishing analysis of the asymptotic properties of the above sequence is well worth reading.
• Very easy:
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25... Give up?
• Not so hard if approached in the right way:
  1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, ... Give up?
• The "RATS" sequence!
  1, 2, 4, 8, 16, 77, 145, 668, 1345, 6677, 13444, 55778, ... Give up?