[seqfan] Re: Another planetary sequence

Thu Sep 15 21:51:19 CEST 2016

> Joerg raises an interesting point: How does one create good OEIS
sequences?

It's not easy to answer! Even determining whether a sequence is good or not
is hard. Alonso has an article here
https://oeis.org/wiki/User:Alonso_del_Arte/Is_this_sequence_interesting
and others have written on the same subject, or on related ones:
http://www.tanyakhovanova.com/Sequences/CreatingNewSequences.html
http://mrob.com/pub/seq/interest.html
http://unsexy-science.blogspot.com/2013/09/random-100-sequences-from-oeis-survey.html

In terms of what I think about when submitting a sequence myself, or on
considering others:
https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Is_this_sequence_interesting
https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Sequences
https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Rule_of_thumb
https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Keywords/aesthetic

Charles Greathouse
Case Western Reserve University

On Thu, Sep 15, 2016 at 3:38 PM, Felix Fröhlich <felix.froe at gmail.com>
wrote:

> Okay, I do accept the decision to reject this sequence. Just let me add
> that sometimes it is difficult to predict (for me at least) whether a
> sequence will be accepted or not. Most sequences are mathematical sequences
> and I think that is good. But there are a number of non-mathematical
> sequences that were still accepted. Don't get me wrong, I am not
> complaining that a sequence of someone else was accepted and mine was not.
> I just mean sometimes it is not obvious to me whether a sequence is or is
> not appropriate. So I rather submit a sequence that gets rejected than not
> submitting a sequence.
>
> Joerg raises an interesting point: How does one create good OEIS sequences?
> I do not know how other contributors work in that regard, but I would love
> to hear more about that. I, for example, often create sequences that have a
> relation to some already existing sequence that interests me, for example
> Wieferich primes (A001220). Or sequences that have some relation to some
> topic or object that I find interesting. I have a large text file on my
> computer where I develop new sequence ideas. That file currently has more
> than 100 entries and most of them are not yet ready to submit (and some
> probably never will be). Most of them are really related to mathematical
> problems, but sometimes something from some other area crosses my mind and
> I make a new section in my text file for it. And then it sits there for a
> while and gets expanded now and then and at some point I may feel that it
> looks somehow "finished" and "okay". That it is "okay" or "interesting" may
> often be subjective.
>
> At that point I start a draft for it and then submit it. I am often quite
> unsure whether a sequence will or will not be accepted. But my personal
> rule of thumb is to submit a sequence rather than not submit, because I
> think withholding a potentially useful addition to the OEIS because of not
> knowing whether it is acceptable or not would be much more harmful than a
> submission that is recycled.
>
> As Charles pointed out above, we discussed the particular sequence in
> question and the long thread above seemed to indicate there is interest in
> such a planetary sequence, so it seemed logical to me to submit it.
>
> Mhm, this has become a rather long message, so I will stop here. I hope
> this gives others a bit of an insight into my motives behind that
> particular submission.
>
> Best regards
> Felix
>
> 2016-09-15 19:55 GMT+02:00 Neil Sloane <njasloane at gmail.com>:
>
> > I guess this is a dispute where I need to step in and resolve things.
> >
> > I agree with Joerg, this is not appropriate for the OEIS.  It is not like
> > it has appeared on an IQ test, which is a legit. reason for including a
> > sequence which would otherwise not be appropriate. It seems contrived and
> > not well-defined (given that the definition of "planet" is a bit
> fuzzy).  I
> > recycled it.
> >
> > Best regards
> > Neil
> >
> > Neil J. A. Sloane, President, OEIS Foundation.
> > 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> > Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> > Phone: 732 828 6098; home page: http://NeilSloane.com
> > Email: njasloane at gmail.com
> >
> >
> > On Thu, Sep 15, 2016 at 1:39 PM, Charles Greathouse <
> > charles.greathouse at case.edu> wrote:
> >
> > > > For an example of "not interesting/contrived" seemingly pertinent to
> > this
> > > discussion:
> > > may I ask what ( https://oeis.org/draft/A276707 ) is supposed to
> convey?
> > > "Number of planet from the Sun that is orbited by the n-th most massive
> > >  natural satellite.
> > >  5, 6, 5, 5, 3, 5, 8, 7, 7, 6, 6, 7, 7, 6, 6"
> > >
> > > This is a sequence I discussed with Felix after this email chain caught
> > my
> > > interest. It's an attempt to do a moon sequence in the right way: no
> > units,
> > > no rounding, and stable over hundreds of millions or billions of years.
> > How
> > > would you do a sequence like this? What do you think of the other
> > > astronomical sequences in the OEIS?
> > > https://oeis.org/wiki/The_multi-faceted_reach_of_the_OEIS#Astronomy
> > >
> > > Charles Greathouse
> > > Case Western Reserve University
> > >
> > > On Thu, Sep 15, 2016 at 1:14 PM, Joerg Arndt <arndt at jjj.de> wrote:
> > >
> > > > Donning my "grumpy old man" hat ...
> > > >
> > > >
> > > > The OEIS is about INTEGER sequences, finite or infinite.
> > > > One can turn real sequences into integer sequences using
> > > > some funny hammer like floor() or round(), this usually
> > > > results in something of little interest.
> > > >
> > > >
> > > > But integer sequences should still have some merit
> > > > (most often/importantly of mathematical nature).
> > > >
> > > > For an example of "not interesting/contrived" seemingly
> > > > pertinent to this discussion:
> > > > may I ask what ( https://oeis.org/draft/A276707 )
> > > > is supposed to convey?
> > > > "Number of planet from the Sun that is orbited by the n-th most
> massive
> > > >  natural satellite.
> > > >  5, 6, 5, 5, 3, 5, 8, 7, 7, 6, 6, 7, 7, 6, 6
> > > > "
> > > > This is just depressing.
> > > > And a maintenance problem.
> > > >
> > > >
> > > > It appears to me that good sequences come up when one works on
> > > > something that does NOT have creating sequences for the OEIS in mind
> > > > in the first place.
> > > >
> > > >
> > > > Best regards,   jj
> > > >
> > > > * Alonso Del Arte <alonso.delarte at gmail.com> [Sep 15. 2016 18:05]:
> > > > > There exist infinitely many sequences, both finite and infinite.
> The
> > > > > problem is to decide which ones are useful and/or interesting
> enough
> > to
> > > > be
> > > > > worth including in this reference work. And then you also have to
> > think
> > > > > about how people are going to look it up to find it.
> > > > >
> > > > > Try the following Wolfram Alpha query: radius of moons of planets
> of
> > > > Solar
> > > > > System.
> > > > >
> > > > > Today, it didn't quite understand that query, and instead gave me
> for
> > > an
> > > > > answer that the planets have an average radius of 15,111 miles, and
> > > this
> > > > > sequence of number of moons: 0, 0, 1, 2, 63 (known), 61 (known), 27
> > > > > (known), 14 (known). If you put in a search for the first four
> terms,
> > > the
> > > > > ones without the disclaimer, you get a couple thousand results.
> > > > >
> > > > > Number of moons certainly looks like a more straightforward
> sequence
> > > than
> > > > > comparing a planet's radius to the radius of its largest moon. But
> > even
> > > > > with this one we have a bunch of asterisks. And could it actually
> be
> > > the
> > > > > case that one of these planets' largest moon is an unknown moon?
> > Sounds
> > > > > unlikely, but it's still enough to cast doubt on both of these
> > > sequences.
> > > > >
> > > > > Al
> > > > >
> > > > > On Wed, Sep 14, 2016 at 4:50 AM, Paul Barry <pbarry at wit.ie> wrote:
> > > > >
> > > > > > Expand  sqrt(1/sqrt(1-12x)).
> > > > > >
> > > > > >
> > > > > > Cheers,
> > > > > >
> > > > > > Paul Barry
> > > > > >
> > > > > > ________________________________
> > > > > > From: SeqFan <seqfan-bounces at list.seqfan.eu> on behalf of Brad
> > Klee
> > > <
> > > > > > bradklee at gmail.com>
> > > > > > Sent: 13 September 2016 23:07:30
> > > > > > To: Sequence Fanatics Discussion list
> > > > > > Subject: [seqfan] Re: Another planetary sequence
> > > > > >
> > > > > > Hi Felix,
> > > > > >
> > > > > > Again, it's a good idea to compute sequences such as this for the
> > > > purpose
> > > > > > of data analysis, and maybe even for a laugh. This one makes
> Mars'
> > > > moons
> > > > > > Phobos and Deimos look pitiful, and another funny observation...
> > > > Jupiter is
> > > > > > #1 when it comes to moons. Ha!
> > > > > >
> > > > > > But think about the "data flood". We probably don't have the
> > manpower
> > > > to
> > > > > > enter every data sequence into the OEIS; though, this would be a
> > > great
> > > > > > "freedom of information act".
> > > > > >
> > > > > > To go back to my original criticism, I think the OEIS is usually
> > > > focused on
> > > > > > infinite series rather than finite sequences.
> > > > > >
> > > > > > There are lots of great series out there, even related to
> planets.
> > > > > >
> > > > > > Consider the "Schwarzschild Solution" , as described in:
> > > > > >
> > > > > > https://en.wikipedia.org/wiki/Two-body_problem_in_general_
> > relativity
> > > .
> > > > > >
> > > > > > There is an effective potential, for which the radius of minimum
> > > > energy can
> > > > > > be found by solving a quadratic equation. Expanding the potential
> > > > around
> > > > > > that r_0 we have something like:
> > > > > >
> > > > > > V(r) = V(r_0) + k*r^2 + . . .
> > > > > >
> > > > > > then we can calculate the scaled zero-radius
> > > > > >
> > > > > > R = (2 * k * m)^(1/4) * r_0
> > > > > >
> > > > > > and substitute into
> > > > > >
> > > > > > 1 - L/R^2
> > > > > >
> > > > > > as in the previous email, the energy-zero-order precession angle
> (
> > up
> > > > to a
> > > > > > factor of 2 pi). We have the exact form for this term, but it's
> > > nothing
> > > > > > pretty to look at. It's much too long to fit into one tiny column
> > of
> > > an
> > > > > > article page. So then, let us expand in powers of x=(G m Sqrt[M
> (m
> > +
> > > > > > M)])/(c L):
> > > > > >
> > > > > > 3*x^2 + (45/2)*x^4 + (405/2)*x^6 + (15795/8)*x^8 +
> (161109/8)*x^10
> > +
> > > > ...
> > > > > >
> > > > > > To compare with standards, take the first term:
> > > > > >
> > > > > > 2 Pi* 3*G^2* m^2 *M *(m + M)/(c* L)^2
> > > > > >
> > > > > > as in the wikipedia article, under the usual assumption (M+m)~M.
> > > > > >
> > > > > > So, how about the numerator / denominator series?
> > > > > >
> > > > > > Numerator: 3, 45, 405, 15795, 161109, 3383289
> > > > > > Denominator: 1, 2, 2, 8, 8, 16
> > > > > >
> > > > > > Denominators look somewhat boring, but numerators are
> interesting,
> > > and
> > > > > >
> > > > > > Not in the OEIS!
> > > > > >
> > > > > > I'll see if I can put this in later when I have double checked
> > > > everything.
> > > > > >
> > > > > > Best Regards,
> > > > > >
> > > > > > Brad
> > > > > >
> > > > > >
> > > > > >
> > > > > > On Tue, Sep 13, 2016 at 8:21 PM, Felix Fröhlich <
> > > felix.froe at gmail.com>
> > > > > > wrote:
> > > > > >
> > > > > > > Thanks for all the further replies. What about the following
> > > > sequence.
> > > > > > Its
> > > > > > > terms are really integers.
> > > > > > >
> > > > > > > Rank of size of largest natural satellite of n-th planet from
> the
> > > Sun
> > > > > > among
> > > > > > > size of all natural satellites of the planets, or 0 if the
> planet
> > > > has no
> > > > > > > natural satellites.
> > > > > > >
> > > > > > > Terms are 0, 0, 5, 67, 1, 2, 8, 7
> > > > > > >
> > > > > > > Of course this sequence may be a bit problematic as well. It is
> > > > still a
> > > > > > > time-dependent sequence, although it probably won't change
> > > > frequently.
> > > > > > Also
> > > > > > > I don't know if that new sequence is that more interesting and
> > it's
> > > > of
> > > > > > > course still finite and relatively short.
> > > > > > >
> > > > > > > Like others in this thread, I have also thought about other
> > > possible
> > > > > > > planetary sequences. What came to my mind was to make a
> sequence
> > > > related
> > > > > > to
> > > > > > > orbital resonances (like the 1:2:4 resonance involving Io,
> Europa
> > > and
> > > > > > > Ganymede), although I am not sure at the moment what that
> > sequence
> > > > could
> > > > > > be
> > > > > > > exactly.
> > > > > > >
> > > > > > > Best regards
> > > > > > > Felix
> > > > > > >
> > > > > > > 2016-09-12 4:58 GMT+02:00 Brad Klee <bradklee at gmail.com>:
> > > > > > >
> > > > > > > > Hi Felix,
> > > > > > > >
> > > > > > > > Yes, the planets are of great interest to everyone, glad you
> > are
> > > > > > > > thinking about it!
> > > > > > > >
> > > > > > > > This sequence is somewhat interesting, but one shortcomming
> for
> > > > OEIS
> > > > > > > > is that it doesn't go on forever, terminates at n=8.
> > > > > > > >
> > > > > > > > Around noon it's a good time to go outside and look up to the
> > > sun.
> > > > > > > > Unless it happens to be hidden, Mercury is somewhere in the
> > sky,
> > > > and
> > > > > > > > then how can you help but to wonder exactly what is going on,
> > > > > > > > physically? With the gravity wave signals coming out of LIGO,
> > > > everyone
> > > > > > > > now believes in Einstein's theory, and maybe they even talk
> > about
> > > > it
> > > > > > > > positively in the beer halls. Time changes everything. But
> some
> > > > things
> > > > > > > > even time has difficulty changing. How many people can do the
> > > > > > > > calculations? For example, perihelion precession? It seems,
> not
> > > so
> > > > > > > > many. Maybe we can use OEIS to help with all of that.
> > > > > > > >
> > > > > > > > We can start with something along the lines of:
> > > > > > > >
> > > > > > > > http://farside.ph.utexas.edu/teaching/336k/Newtonhtml/
> > > node115.html
> > > > > > > >
> > > > > > > > but it's still difficult even if you know squares of the
> > > > y-intercepts
> > > > > > > > of the Legendre polynomials. This presentation is
> confounding,
> > at
> > > > > > > > least to me. Luckily I did figure out another _Big Equation_
> > that
> > > > > > > > works just as well __on all planetary data__.
> > > > > > > >
> > > > > > > > If you read through the following and referenced articles
> > > therein:
> > > > > > > >
> > > > > > > > http://oeis.org/search?q=pendulum+klee&language=
> > > english&go=Search
> > > > > > > >
> > > > > > > > you should get a decent idea as to how to find the exact
> > > solutions
> > > > of
> > > > > > > > 1-D oscillation problems in Classical Mechanics as expansions
> > in
> > > an
> > > > > > > > energy parameter, which leads along one branch to the famous
> > > > Jacobian
> > > > > > > > Elliptic integrals.
> > > > > > > >
> > > > > > > > In a pseudo-potential formulation that Gauss and later
> > > Fitzpatrick
> > > > > > > > suggest for a first analysis of planetary perihelion
> > precession,
> > > > you
> > > > > > > > can essentially reduce the two-dimensional orbital motion to
> > one
> > > > > > > > dimension ( by conservation of angular momentum ), and do
> more
> > > > exact
> > > > > > > > solving along similar lines. This time not for the period,
> but
> > > for
> > > > the
> > > > > > > > precession rate.
> > > > > > > >
> > > > > > > > Using Mathematica to expedite multiplication and addition, I
> > > wrote
> > > > the
> > > > > > > > following equation two days ago:
> > > > > > > >
> > > > > > > > 1-L/r^2
> > > > > > > > +a*( (-3/2)*L/r^4 + (-3)*L*v[3]/r^3 + (-15/4)*L*v[3]^2/r^2 +
> > > > > > > > (3/2)*L*v[4]/r^2 )
> > > > > > > > + a^2 *((-15/8 )*L / r^6 + (-15/2)* L * v[3]/r^5 + (-315/16)*
> > L*
> > > > > > > > v[3]^2/r^4 + (-315/8)* L *v[3]^3/ r^3 + (-3465/64)* L *
> v[3]^4/
> > > > r^2 +
> > > > > > > > ( 45/8 ) * L *v[4]/r^4 + (105/4)* L *v[3] *v[4]/ r^3 +
> > (945/16)*
> > > > > > > > L*v[3]^2 *v[4]/ r^2 + (-105/16)* L *v[4]^2/r^2 +(- 15/4)* L
> > > > *v[5]/r^3
> > > > > > > > + (-105/8)* L *v[3]* v[5]/ r^2 + (15/8)* L *v[6]/r^2 )
> > > > > > > > +a^3 *( ...) + ... higher order in a ...
> > > > > > > >
> > > > > > > > which describes perihelion precession in a general isotropic
> > > > central
> > > > > > > > potential in terms of potential expansion coefficients v[i],
> > > > angular
> > > > > > > > momentum "L", radius "r", and energy "a". As a check, for
> > values
> > > > of (
> > > > > > > > v[i], r ) determined from coulomb potential the whole
> expansion
> > > > goes
> > > > > > > > to zero for any pair of (L,a). The coulomb solutions are
> Kepler
> > > > > > > > Ellipses fixed in space, i.e. no precession.
> > > > > > > >
> > > > > > > > It may seem silly to have such a big equation that gets even
> > > > bigger,
> > > > > > > > but lets just look at the results. We have to use potential
> > > theory
> > > > to
> > > > > > > > calculate values for
> > > > > > > >
> > > > > > > > {a,r,v[1],v[2],v[3],v[4],v[5], ... }
> > > > > > > >
> > > > > > > > by transforming the data at
> > > > > > > >
> > > > > > > > http://nssdc.gsfc.nasa.gov/planetary/factsheet/
> > > > > > > >
> > > > > > > > This is the only hard part, but doesn't take too long.
> Plugging
> > > in
> > > > the
> > > > > > > > numbers for Mercury, and scaling to arcseconds per year, we
> > get,
> > > > term
> > > > > > > > by term:
> > > > > > > >
> > > > > > > > {4.67068, 0.747725, 0.10366, 0.0149433, 0.0022804}
> > > > > > > >
> > > > > > > > with folded sum
> > > > > > > >
> > > > > > > > {4.67068, 5.41841, 5.52207, 5.53701, 5.53929}
> > > > > > > >
> > > > > > > > which converges to a value near the 5.5 reported by
> Fitzpatrick
> > > in
> > > > his
> > > > > > > > table 2. For other planets the convergence is much faster,
> and
> > > the
> > > > > > > > second order equation here should do just fine. The
> > > > pseudo-potential
> > > > > > > > for Mercury is the most significantly different from
> quadratic
> > > > > > > > harmonic, so higher terms are noticeable.
> > > > > > > >
> > > > > > > > The interesting part of this analysis is that energy
> parameter
> > > "a"
> > > > > > > > takes into account the time-changing radius of planetary
> > orbits,
> > > > > > > > whereas I cannot see any time-changing radius of the orbits
> in
> > > > > > > > Fitzpatrick's equations! In our approach there is a
> > contribution
> > > > > > > > independent of "a", but notice that for Mercury values
> > > > > > > >
> > > > > > > > 4.67068 =/= 5.5 ( Fitzpatrick value )
> > > > > > > >
> > > > > > > > and
> > > > > > > >
> > > > > > > > 4.6708 =/= 5.32 ( https://en.wikipedia.org/wiki/
> > > > > > > > Tests_of_general_relativity )
> > > > > > > >
> > > > > > > > so we need to include the part that depends on how high in
> the
> > > > > > > > effective potential the system is to get nearer to the
> correct
> > > > answer.
> > > > > > > > How does Fitzpatrick get 5.5 with his method? Still wondering
> > > about
> > > > > > > > that, especially if his expansion does indeed ignore terms
> > > > depending
> > > > > > > > on the energy of the orbit. ( Maybe a closed form summation
> > along
> > > > "a"
> > > > > > > > ? Even then, does his expansion have cross terms such as  L
> > *v[3]
> > > > > > > > *v[4]/ r^3 ? ).
> > > > > > > >
> > > > > > > > You may already notice that there is a fractional sequence of
> > > > > > > > coefficients in the big precession equation. We can ( and
> will
> > )
> > > > put
> > > > > > > > this into the OEIS, as an irregular triangular. Although
> there
> > is
> > > > an
> > > > > > > > infinite number of v[i] variables, there is a natural grading
> > in
> > > > the
> > > > > > > > ring generated by all possible products of the variables,
> which
> > > can
> > > > > > > > already be seen in the first few terms. If you have term
> > > > > > > >
> > > > > > > > v[i]^p_i * v[j]^p_j *..../r^p_r
> > > > > > > >
> > > > > > > > Then the exponents must satisfy
> > > > > > > >
> > > > > > > > ( i - 2 )*p_i + ( j - 2 )*p_j + ... p_r = 2(N+1)
> > > > > > > >
> > > > > > > > so every row is finite. We'll just need to do a Lexicographic
> > > > ordering
> > > > > > > > of the variables and exponents, and accept that the triangle
> > will
> > > > > > > > probably have lots of zeros. While we're on the subject:
> > > > > > > >
> > > > > > > > ** Does anyone know of a canonical form in OEIS for such a
> > > > sequence?
> > > > > > > > Or a sequence along similar lines? **
> > > > > > > >
> > > > > > > > Anyways it's a good place to leave off. This idea may yet be
> > > > > > > > confusing, but will be much easier to understand once its
> been
> > > > > > > > published through a few different venues including the OEIS.
> > > > > > > >
> > > > > > > > Thanks for writing, happy planet gazing!
> > > > > > > >
> > > > > > > >
> > > > > > > > <~~~?~~~~~~
> > > > > > > >
> > > > > > > >               ?
> > > > > > > > ~~~~~~~~?~~~~~~~~~~~>
> > > > > > > >
> > > > > > > >           ~~?~~~>
> > > > > > > >
> > > > > > > > Watch out for troublesome syzygies!
> > > > > > > >
> > > > > > > >                          ~?~>
> > > > > > > >
> > > > > > > >
> > > > > > > > Best regards,
> > > > > > > >
> > > > > > > > Brad
> > > > > > > >
> > > > > > > >
> > > > > > > >
> > > > > > > >
> > > > > > > >
> > > > > > > >
> > > > > > > > > On Sep 11, 2016, at 11:49 AM, Felix Fröhlich <
> > > > felix.froe at gmail.com>
> > > > > > > > wrote:
> > > > > > > > >
> > > > > > > > > Dear sequence fans
> > > > > > > > >
> > > > > > > > > I thought about the following sequence:
> > > > > > > > >
> > > > > > > > > Ratio of radius of n-th planet (under the current
> > > IAU-definition
> > > > of
> > > > > > > > planet)
> > > > > > > > > from the Sun to mean radius of its largest natural
> satellite,
> > > > rounded
> > > > > > > to
> > > > > > > > > the nearest integer, or 0 if the planet has no natural
> > > satellite.
> > > > > > > > >
> > > > > > > > > a(1)-a(8) are 0, 0, 4, 308, 27, 23, 32, 18
> > > > > > > > >
> > > > > > > > > The true ratios are of course not integer values, but the
> > terms
> > > > give
> > > > > > an
> > > > > > > > > idea of how large the largest moons of the planets are
> > compared
> > > > to
> > > > > > the
> > > > > > > > > planets themselves (i.e. these values are still useful for
> > > > > > comparison,
> > > > > > > > even
> > > > > > > > > when rounded to integers in my opinion). The closer the
> value
> > > is
> > > > to
> > > > > > 1,
> > > > > > > > the
> > > > > > > > > larger the largest moon is relative to its planet. The
> value
> > > for
> > > > > > > Earth's
> > > > > > > > > moon is relatively small, meaning the Moon is large
> relative
> > to
> > > > > > Earth.
> > > > > > > > >
> > > > > > > > > There are already a number of sequences related to the
> > planets
> > > > in the
> > > > > > > > OEIS,
> > > > > > > > > but I would like to hear the opinion of other contributors
> > > and/or
> > > > > > some
> > > > > > > of
> > > > > > > > > the editors before submitting this, mainly because I think
> > the
> > > > > > sequence
> > > > > > > > > will likely be rejected.
> > > > > > > > >
> > > > > > > > > It is probably a "dumb" sequence, not really mathematically
> > > > > > > significant,
> > > > > > > > > but sometimes such sequences are still enjoyable.
> > > > > > > > >
> > > > > > > > > Best regards
> > > > > > > > > Felix
> > > > > > > > >
> > > > > > > > > --
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> > > > > > > > --
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> > > > > --
> > > > > Alonso del Arte
> > > > > Author at SmashWords.com
> > > > > <https://www.smashwords.com/profile/view/AlonsoDelarte>
> > > > > Musician at ReverbNation.com <http://www.reverbnation.com/
> > > alonsodelarte>
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