Universal Sequence Index (USI)?

[Jon Awbrey]

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SeqFans,

Several recent notes have given me flashbacks to this old theme,
which I just realized how to express in more contemporary terms.
For some reason Marc's note makes me think of Conway's Fractran.

EIS is a resource that I will call "Lot_X", or "L_X" for short,
X being an index value to be named later.  I am tempted to let
X equal something between 1 and 7^2, but I do not know yet how
much of an initial segment might be needed for other resources.

I do not have a complete idea here, but here are a couple
of fragmentary ideas that may form the ingredients of one:

Ingredient 1.

If L_X^(L_A061396) = L_X^(L_A061396^1) is my initial sequence,
then the expression "L_X^(L_A061396^n)" might serve to denote
A061396's n-th element, although I don't want to fix this too
rigidly right yet as other ways of doing this may work better.

Ingredient 2.

Now, riffs are constituted in such a way as to yield a code value v(f)
for each finite partial function f : M -> M, where M = 1, 2, 3, 4, ...
Is there a good way of using this natural scheme to specify a sequence
in terms of references to other sequences, assuming that we can design
all the details of de/referencing to work out as elegantly as possible?
Exercise for the reader.  I'll think about it a bit and get back to you.

Jon Awbrey

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Marc LeBrun wrote:
> 
> Like a moth to a candle, David Wilson's note about Collatz being
> 2-automatic reminded me of something ... I've wondered if anyone
> else has ever explored anything along these lines?
> 
> Obviously since 3n+1 is even when n is odd, we can just as well fold in the
> inevitable halving step, getting f(n) defined by
>
>    2k   --> k
>    2k+1 --> 3k+2
> 
> The nearly trivial observation is just that f(n) is very similar
> to the "rotate right" function r(n) = A038572:
> 
> f(n):  0 2 1 5 2 8 3 11 4 14 5 17 6 20 7 23 8 26 9 29 10 32 11 35 12 38 13
>        41 14 44 15 47 16 ...
> r(n):  0 1 1 3 2 6 3  7 4 12 5 13 6 14 7 15 8 24 9 25 10 26 11 27 12 28 13
>        29 14 30 15 31 16 ...
> diff:  0 1 0 2 0 2 0  4 0  2 0  4 0  6 0  8 0  2 0  4  0  6  0  8  0 10  0
>        12  0 14  0 16  0 ...
> 
> These even positive differences are just twice the sequence
> 
>         1, 1 2, 1 2 3 4, 1 2 3 4 5 6 7 8, 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...
> 
> Anyway, while f(n) apparently iterates chaotically to 1, "destroying" all the
> information originally in n, in contrast r(n) preserves the 1 bits, and so fixes
> onto the points 2^k - 1.
> 
> This makes it smell to me like information-theoretic approaches to
> Collatz, such as this k-automatic stuff, might indeed be very fruitful.
> 
> (Or at least there's some simple sequences I owe the EIS... when I'm more awake!<;-)

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