[seqfan] Re: Glass worms

[Joseph S. Myers]

On Mon, 7 Sep 2009, N. J. A. Sloane wrote:

> Dear Sequence Fans, especially Eric Angelini,
> 
> I like vers de verres a lot, and I'm adding two sequences 
> from Jacques Tramu's work, A151986, A151987.
> 
> (This refers to postings from March 9 2009,
> which I finally had time to read)

The message
http://list.seqfan.eu/pipermail/seqfan/2009-March/001066.html
giving the figures for A151986 says "not exhaustive search", which I 
interpret as indicating that the values may be conjectural (maximum found, 
rather than known to be maximum possible).  My program (finding all 
possible cycles then searching backwards from those for long paths leading 
to a cycle) agrees with the given figures up to 10 glasses, but I get 64 
not 63 for 11.  The following are the figures and (not necessarily unique) 
worst examples according to my program; can someone at least confirm these 
examples do give the indicated figures as lower bounds (so I'm 
interpreting the problem in the intended way)?

1 glasses: max 1: 0
2 glasses: max 2: 0 1
3 glasses: max 4: 0 1 1
4 glasses: max 7: 0 0 2 1
5 glasses: max 10: 0 0 0 3 1
6 glasses: max 13: 0 0 0 3 0 2
7 glasses: max 20: 0 0 0 6 0 3 1
8 glasses: max 22: 0 0 0 0 0 4 2 1
9 glasses: max 42: 0 0 0 0 0 5 0 6 3
10 glasses: max 43: 0 0 0 0 0 0 5 0 6 3
11 glasses: max 64: 0 0 0 0 0 8 0 5 3 0 2
12 glasses: max 65: 0 0 0 0 0 0 8 0 5 3 0 2
13 glasses: max 146: 0 0 0 0 0 12 0 8 0 5 3 0 2
14 glasses: max 147: 0 0 0 0 0 0 12 0 8 0 5 3 0 2
15 glasses: max 203: 0 0 0 0 0 0 0 0 0 8 6 12 3 0 5
16 glasses: max 239: 0 0 0 0 0 0 0 12 0 9 6 0 5 0 3 1
17 glasses: max 412: 0 0 0 0 0 0 0 0 0 0 9 14 12 0 4 2 1
18 glasses: max 413: 0 0 0 0 0 0 0 0 0 0 0 9 14 12 0 4 2 1
19 glasses: max 414: 0 0 0 0 0 0 0 0 0 0 0 0 9 14 12 0 4 2 1
20 glasses: max 415: 0 0 0 0 0 0 0 0 0 0 0 0 0 9 14 12 0 4 2 1

I think the offset of A151987 (which my program does not compute) should 
be 0 not 1.  The link from those sequences to "Discussion about Glass 
Worms" gives an error, "You don't have permission to access 
/~oeis/sequences/ADATA/a151986.txt on this server.".

-- 
Joseph S. Myers
jsm at polyomino.org.uk