Hi Dan, Clark Kimberling seems to be the reigning expert on fractal sequences and has published several of them in OEIS. His page here gives a little information: http://faculty.evansville.edu/ck6/integer/fractals.html MathWorld gives a bit more detail: http://mathworld.wolfram.com/FractalSequence.html A sequence can be considered fractal if it is transformed back to itself under upper trimming or lower trimming. Particular examples of those types of fractal sequences are signature sequences: http://mathworld.wolfram.com/SignatureSequence.html More generally, a fractal sequence is any infinite sequence that, in some fashion, contains infinite copies of itself. The upper- and lower-trimmed fractals sequences satisfy that notion in the respect that they can be trimmed infinitely and always come back to the same sequence. Another notion is exhibited by this sequence (A000265<http://www.research.att.com/projects/OEIS?Anum=A000265> in OEIS): 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 1, 17, 9, 19, 5, ... If you delete all the elements with odd-numbered indices (which in this case amounts to upper trimming), then you recover the same sequence. Or, the Thue-Morse sequence (A010060<http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A010060> ): 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, ... Delete the elements with even-numbered indices and you have the same sequence. Or, delete every second block of 2 elements or every second block of 4, or every second block of 2^n elements. The Fibonacci "rabbit" sequence (A005614<http://www.research.att.com/projects/OEIS?Anum=A005614> ) 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, ... Is considered fractal because the string replacement 1, 0 -> 1 and 1 -> 0 turns the original sequence into itself. Since the exact manner in which a sequence can contain copies of itself varies, I don't know that there is an exact definition of a fractal sequence. Kerry On 12/4/05, dasimov@earthlink.net <dasimov@earthlink.net> wrote:

Kerry,

You've said some interesting things about fractal sequences, like how to detect them.

But what is their underlying definition?

Thanks,

Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun