Sequences

[geodave]

...does it still work for a random string of numbers? I'm guessing there has to be SOME pair of numbers that divides in such a way (rational or otherwise) as to produce the string of numbers, but...

Yes. I don't want to totally geek out here (too late!), but Klein (I think it was him) basically proved that there is a mathematical formula for ANY finite sequence of numbers. If you think about it, that kind of makes sense. Anyway, Gödel used it in his incompleteness proof, which is one of the greatest things ever that no one knows about.

You can always use a polynomial approximation to get a valid "next number" in a sequence. You just have to make the power high enough to give you enough unknowns and enough formulas. (Or you can let a computer solve the matrix for you.)

OR, and this is incredibly math-geeky, you can use repeated differences to come up with a next number:

1,3,7,24,46,111,.....

2,4,17,18,65

2,13,1,43

11,-12,42

-23,54

77

77,77

-23,54,131

11,-12,42,173

2,13,1,43,216

2,4,17,18,65,281

1,3,7,24,46,111,392

Which, in theory, is a six-degree polynomial. I'm not going to bore you with that derivation.