Identification of fractional sequences

[Simon Plouffe]

This is a classic case, (if I may),

you should use GFUN (maple) or the equivalent
on Mathematica, with these tools you can
manipulate sequences as if they were exponential
or ordinary gen. functions and from there easily
identify any possible sequence.

the command are <listtoseries> , etc.

type ?gfun on a maple session

simon plouffe



Leroy Quet <qq-quet at mindspring.com> wrote:
:hv at crypt.org wrote:
:>...
:>I get (3, 3): 3 3 0 2 1 2 2 0 3 3 0 4 1 0 2 4 0 2 5 0 1 3 2 1 5 0 2 7 0 1 5 0
:>3 6 0 1 6 1 1 7 0 2 8 0 1 9 0 1 10 0 1 11 0 1 12 [...]
:>
:>(5, 5): 5 5 0 2 1 0 1 2 0 2 3 2 2 0 5 3 0 2 6 0 1 3 0 3 4 0 1 4 3 0 5 4 0 3 6
:>0 2 7 0 1 5 2 1 6 1 1 8 0 1 9 0 1 10 0 1 11 0 1 12 [...]
:>
:>Giving:
:>  0 1 2 3 4 5
:>0 . . * * * *
:>1 . . . * * *
:>2 . . . * * *
:>3 * * * . * *
:>4 * * * * * *
:>5 * * * * * .
:>
:>I think there may be some interesting maths involved in characterising
:>the behaviour of the 3-valued function f(a(0), a(1), n), I agree that
:>there doesn't currently seem much of an interesting sequence here.
:>
:>Hugo
:
:I was hoping for some interesting pattern of .'s and *'s when the above 
:plot is taken to many more terms. But I conjecture, sadly, that the only 
:.'s in the infinite grid are those represented above, the rest of the 
:grid-squares being of only *'s.
:
:(Again: "*" represents a (a(0),a(1)) which leads to a sequence which 
:repeats after some point. "." represents a (a(0),a(1)) which leads to a 
:sequence which contains arbitrarily large terms.)

If you allow more than 2 initial terms, you can maybe get more interesting

It also raises the question in the bounded cases of what sequences are
possible in the repeating part - I suspect it is exactly those loops which
do not include any of the values of their pairwise sums, but it might not
be quite so simple. (For example, trivially, the initial values
[9 9 9 9 10 10 10 7 7 7 7 7 7 3 6 4] yield the loop [3 6 4].)

Hugo



And the hunt continues:


A108163 and A108171
A118713 and A118756
A127554 and A127555
A050279 and A096764
A069582 and A089375
A023237 and A105434
A106109 and A106149


possible duplicates:

A087316 and A113171
A085505 and A086863 (examples mostly match, is one of these sequences
wrong?)
A076265 and A122149 (same definition, different contexts, should they be
merged?)
A083772 and A093441
A086395 and A111008
A016081 and A119032