[SeqFan]: Repeated iterations of INVERT starting from A019590 ?

[Antti Karttunen]

Cheers,

concerning the repeated invocations of INVERT-transform
(see http://www.research.att.com/~njas/doc/eigen.ps or 
http://www.research.att.com/~njas/doc/eigen.pdf
for the exact definition of INVERT) to which I already
delved 2 and half years ago:

John Layman wrote Dec 8, 2003:

 >Subject: Some INVERT transforms.
 >From: John Layman <layman at calvin.math.vt.edu>
 >To: seqfan at ext.jussieu.fr
 >Date:   Mon, 8 Dec 2003 11:33:58 -0500 (EST)
 >Cc: layman at calvin.math.vt.edu (John Layman)
 >Message-Id: <20031208163403Z10491-608+64 at calvin.math.vt.edu>

 >Antti Karttunen wrote (Dec 8, 2003): ...
 >>could you compute for me the following transforms:
 >>INVERT([1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]);
...

 >John: Here are some results that I obtained using my own ISAP (Integer 
Sequence Analysis Program),
 >written in Pascal (US denotes the entered USer sequence):

 >US =1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (25)
 >(INV)US = 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 
6765 10946 17711 28657 46368 75025 121393 (25)

I am now proposing that if we start from INVERT(A019590),
and then repeatedly apply INVERT(RIGHT(...)) to the result,
(here RIGHT will just insert 1 to the front of sequence)
the resulting sequences will converge towards A000108, Catalan numbers.
This purely on combinatorial grounds, as a side product of "Gatomorphology".
Note that A000108 is an eigen-sequence regarding to this
operation, i.e. it stays exactly same after INVERT(RIGHT(...)).
(Stronger conjecture is that such an iteration procedure, starting from
_almost any_ sequence, eventually converges towards A000108.)


I.e. INVERT(A019590) produces Fibonacci numbers beginning from A000045(2),
and when we apply INVERT(RIGHT(...)) to them, i.e.
INVERT([1,1,2,3,5,8,13,21,34,55,89,...])
we get Pell numbers, A000129
1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741,
and again, prepending one, and taking INVERT:
INVERT([1, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, ...]
produces what?

Could somebody compute that, and continue few iterations more?
(Remember the prepended 1 and be careful with those indices,
as INVERT-transform doesn't/shouldn't actually work on any 0-indiced terms!)

This might make a nice table, with some other nice properties as well.

Also, if anybody knows a pointer to a free INVERT-procedure/algorithm,
then let me know, then I can compute it by myself.


Terveisin,

Antti