Anyone for Twopins? In the dust of A112088.

[Rainer Rosenthal]

The exploration of A112088 is still under work,
but here is something along the way, which is
interesting. Sequence A103609 belongs to the
"Anyone for Twopins" family or at least it is
quite probable so because of the authors reference
to Fibonacci Numbers.

The Twopins sequences already contained in the OEIS
(and mentioned as such) are:

A005682: 1, 2, 4, 8, 15, 28, 51, 92, 165, 294, 522, ...
A005683: 1, 2, 3, 5,  8, 13, 22, 37,  63, 108, 186, ...
A005684: 1, 2, 4, 6, 11, 18, 32, 52,  88, 142, 236, ...
A005685: 1, 2, 3, 5,  7, 11, 16, 26,  40,  65, 101, ...
A005686: 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 9, 12, 14, ...
A005687: 1, 2, 4, 6,  9, 14, 22, 36,  57,  90, 139, ...
A005688: 1, 2, 3, 5,  7, 10, 14, 20,  30,  45,  69, ...
A005689: 1, 2, 4, 7, 11, 16, 22, 30,  42,  61,  91, ...
A005690: 1, 2, 4, 6,  9, 12, 18, 26,  41,  62,  96, ...
A005691: 1, 2, 3, 5,  7, 10, 13, 18,  24,  35,  50, ...

As a proud owner (since today) of
R. K. Guy, ``Anyone for Twopins?,''
in D. A. Klarner, editor,
The Mathematical Gardner.
Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15,

I checked some of the sequences on p. 11, among them:
1,2,2,3,3,5,5,8,8,13,13,21,21,34,34,55,55,89

and - alas! - it is Roger Bagula's sequence A103609
with formula  a(n) = a(n-2)+a(n-4).

I am not sure if a comment would be correct, pointing
to the Twopins family, because I don't have any proof
but only a "strong feeling" that it might be correct.

Ralf Stephan may be specially interested as I conclude
from his many remarks in the Twopins sequences.

---

Checking now in a more systematic way thru pp. 11 ff:
1, 2, 3, 5, 7, 10, 13, 18, 24, 35, ...  is A005251
1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28  is A000931
1, 1, 1, 1, 2, 4, 7, 11, 17, 27, 44, 72 is A005252 (*)
1, 2, 4, 7, 11, 16, 23, 34, 52, 81, 126 is A005253
... 19, 19, 28, 28, 41, 41, 60, 60, 88  is A108104 (**)
... 16, 22, 29, 38, 51, 71, 102, 149,   is A098574
... 20,25,29,35,42 (page 12)            not in the OEIS
... 46,68,102 (page 12)                 not in the OEIS
... 40,56,81,116 (page 12)              not in the OEIS
... 29,37,47,61,82 (page 12)            not in the OEIS
... 19,19,26,26,36,36 (page 12)         not in the OEIS
... 27,36,52,74,109 (page 12)           not in the OEIS
... 17,21,27,34,46 (page 12)            not in the OEIS
... 29,37,46,57,72 (page 13)            not in the OEIS
... 33,37,44,51,59 (page 13)            not in the OEIS
... 58,81,116,164 (page 13)             not in the OEIS
... 40,52,69,95 (page 13)               not in the OEIS
... 68,84,107,141 (page 13)             not in the OEIS
... 20,20,26,26,34,34 (page 13)         not in the OEIS
... 65,88,124 (page 13)                 not in the OEIS
... 59,76,103 (page 13)                 not in the OEIS

(*) This is especially interesting, isn't it?
(**) Another sequence from Roger Bagula!

Best regards,
Rainer Rosenthal
r.rosenthal at web.de

P.S. I am sending to both SeqFan addresses, because I
do not trust the new one :-)