[seqfan] Sequence that contains A028242

jnthn stdhr jstdhr at gmail.com
Wed Oct 4 20:49:43 CEST 2017

```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
```