[seqfan] Re: new dragon conjecture

[brad klee]

> I didn't try to prove it, and I don't know how difficult it would be. Does
> anyone around here want to take a crack at it? Is A279645 truly the
> "spigot differences" of A014577?

At the risk of sounding like a St. Petersburg madman by quoting and
responding to myself while passerbys rush to the coffee shop --

I sketched a proof last night. It wasn't too much to ask myself for,
see notes below. The upshot is a new (?) definition for the regular
paper folding sequence (or dragon sequence if you like):

anew[n_] := Module[
{ind = IntegerDigits[n, 2], vals, pos},
vals = Mod[Plus[Prepend[Most[ind], 0], ind], 2];
pos = First[FirstPosition[Reverse[ind], 1]];
vals[[-pos]] ]

It's not more efficient than what we already have, so not critical
to include. If there's a similar definition for the new 4-ary sequence
that would be interesting to write out, and would perhaps make
it easier to enter the data into OEIS.

If the proofs are not feasible, does anyone care to give a more
concise definition of the 4-ary sequence with these as the first
few terms? Here they are again:

https://www.youtube.com/watch?v=5mnYCWSNA6g
(and the article from yesterday has spigot code)

If you look more into it, the neopixel arrays can get quite
large and display a lot of data per instant. However for this
particular use case, four pixels made a nice MWE:

→ Like the digits of pi, the sequence never falls into a periodic

repeating pattern.

→ Unlike the digits of pi, the sequence will never display four
of the same value in a row.

Thanks for your time!

--Brad

PROOF SKETCH

→ Incrementing the bit vector adds one then carries
→ On each carry, a digit changes
→ When carry stops, change stops
→ For more significant digits where nothing changes,
not (n XOR (n shift 1)) also does not change
→ Long carry means, many ones changing to
many zeros
→ many oens changed to many zeros does not
change not (n XOR (n shift 1))
→ Where the carry stops, a zero changes to one
→ Where the zero used to be the one is, and
not (n XOR (n shift 1)) must change there
→ The change of 0 to 1 or 1 to 0 depends
upon the more significant digit, which did not
change.
[♪]