[seqfan] Re: Sequence that contains A028242

[M. F. Hasler]

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