[seqfan] Re: Sequence that contains A028242

[Marc LeBrun]

Yes, you should submit this sequence!


> On Oct 4, 2017, at 11:49 AM, jnthn stdhr <jstdhr at gmail.com> wrote:
> 
> Hello seqfans.
> 
> 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) }
> 
> The odd terms appear to produce A028242 (Follow n+1 by n. Also
> (essentially) Molien series of 2-dimensional quaternion group Q_8).
> 
> With the lookup of "0 1 1 0 1 2 1 1 1 3 1 2 1 4 1 3 1 5"
> superseeker suggests the following generating functions:
> 
> (1)
> 
> "Generating function(s) and type(s) are:
> 
> 2 3 4 5 6 [[RootOf(y - _Z + (-y - 1) _Z + _Z - y _Z - _Z + (y + 1) _Z )],
> revogf]
> 
> revogf = reversion of ordinary generating function"
> 
> (2)
> 
> "TRY "GUESSS", HARM DERKSEN'S PROGRAM FOR GUESSING A GENERATING FUNCTION
> FOR A SEQUENCE.
> 
> 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] "
> 
> The next six terms in (2) are correct.  I don't understand (1).
> 
> Since it appears this is just A028242 with ones inserted between terms,
> should I bother to submit this sequence?
> 
> -Jonathan
> 
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