[seqfan] Re: Sequence that contains A028242

[jnthn stdhr]

Ok.  One yes, two no.  I will use the allocated A-number for something else.
On Oct 5, 2017 7:34 AM, "M. F. Hasler" <oeis at hasler.fr> wrote:

> On Wed, Oct 4, 2017 at 2:49 PM, jnthn stdhr wrote:
>
> > The sequence 0,1,1,0,1,2,1,1,1,3,... is not in the database.
>
> It is produced by the following:
> > a(0) = 0, a(1) = 1, a(2) = 1, a(3) = 0;  a(n) = { if n is even, a(n - 2);
> > if n is odd, a(n - 3) + a(n - 4) }
> >
>
> Well,  a(n) = a(n-2) for n even would mean that it's a constant sequence,
> but you set (somehow "inconsistently") a(0)=0 but a(2)=1.
> Let's accept this irregularity, then a(2k) = 1 for all k >= 1, and I think
> it would be more interesting to consider only the sequence of odd-indexed
> terms for which the rule is actually (starting with n = 2k+1 = 5 you never
> need the a(0)=0 so you can substitute a(n-3)=a(even)=1) :
> b(k) = a(2k+1) = 1 + a(2k-3) = 1 + b(k-2),
> which is a quite simple sequence : (1,2,3,4,...) interleaved with
> (0,1,2,3....), or b(2m) = m+1, b(2m+1) = m.
>
>
> The odd terms appear to produce A028242 (...)
> >
>
> This is indeed A028242,
>
>
> > Guesss suggests that the generating function F(x) may satisfy the
> following
> > algebraic or differential equation:
> > x^6-x^5+x^3-x^2-x+(x^6-x^2-x^4+1)*F(x) = 0
>
> If this is correct the next 6 numbers in the sequence are:
>
> [1, 4, 1, 6, 1, 5] "
>
>
>
> and (thus of course) agrees with the next terms you got with the G.F.,
> but there is not really any fancy math required to find that...
>
> Since it appears this is just A028242 with ones inserted between terms,
> > should I bother to submit this sequence?
> >
>
> Not sure whether it is really worth submitting this sequence, IMHO it is a
> "diluted" A028242 (which itself is not very "dense");
> anyone trying to find a sequence with every second term equal to 1 would
> discard these 1's and would look for the other terms and find the seqence
> A028242 at once.
> Of course one could argue that /any/ sequence should be in the
> encyclopedia, but then we should start with A0028242 interleaved with 0's,
> and do the same for all the core sequences, and only then proceed to
> interleave all sequences with 1's, and/or larger values.
> (If the sequence would appear as is in some "interesting" context or as
> solution to a nontrivial recurrence, then the situation would be different,
> but here the recurrence is trivial in the sense of being decoupled between
> odd and even (all 1's) terms.
>
> Maximilian
>
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