derivative of n

[Marc Le Brun]

>=N. J. A. Sloane
>a new class of sequences.

Very pretty!

But, actually, aren't these a subclass of one-dimensional cellular automata
(some even with so-called "totalistic" rules, no less)?

>The derivative of a binary sequence is obtained by adding the bits
>in pairs:   the derivative of 1 1 1 0 1 0 0 0 1 1 is
>                               0 0 1 1 1 0 0 1 0

That is, approximately N XOR (N shift 1) right?

>The periodic derivative includes the sum of the last bits:
>1 1 1 0 1 0 0 0 1 0 has periodic derivative
>0 0 1 1 1 0 0 1 1 1

That is, substituting "rotate" for "shift" in the above expression, where
the "word width" is defined by the MSB of N, right?

This just inspired me to submit the even simpler
  0, 1, 1, 3, 2, 6, 3, 7, 4, 12, 5, 13, 6, 14, 7, 15, 8, 24, 9, 25, 10, 26,...
that is, N rotated 1 place right.  I was surpised that this was missing!
It's "dual", rotate left, is A6257 (NJAS: could you cross link this with
the above new sequence?)

Because the "word width" decreases whenever 0s "bubble" into the MSBs,
these functions have fixed-points of the form 2^k-1, specifically, the
smallest number with the same number of one bits as n:
  0, 1, 1, 3, 1, 3, 3, 7, 1, 3, 3, 7, 3, 7, 7, 15, 1, 3, 3, 7, 3, 7, 7, 15,
3...
which I'm afraid I haven't the time to check/submit right now (due to all
the cross links I might have to chase down to do it right), along with a
host of variations.

I guess I'll have to return to this later, if no one else does it first...

Thanks!