# [seqfan] Re: Need help with puzzling sequence A118980

Neil Sloane njasloane at gmail.com
Tue Aug 13 00:33:35 CEST 2019

```By looking more carefully at Richard Mathar's work from 2011, I was able to
straighten out the mess.
There is a new sequence A309220 that replaces the missing "A118979",
and A118980 is now well-defined and edited. Ditto for A118981.

The key was Richard Mathar's 2011 conjecture that A104509 drives the bus.

I took A104509 as the starting point, and defined the other sequences
in terms of it.  A118980 and A309220 have Maple code, based on Robert
Israel's  Maple code for A104509.

On Mon, Aug 12, 2019 at 6:21 PM Fred Lunnon <fred.lunnon at gmail.com> wrote:

>   Apologies for attack of fat finger syndrome ...
>
> What I intended to say was that I suspect the author made an elementary
> error right at the top of the table: it should have commenced
>
> 1
>
> 2  1
>
> 5  5  2
>
> ...
>
> Then the first column and final diagonals satisfy
>
> T(n,0)  =  2 T(n-1,0) + T(n-2,0) ;
>
> T(n,n-1)  =  n! (n+3)/2 ;
>
> T(n,n)  =  n!  ;
>
> Without more consistent data we're still snookered at this point ...
>
> WFL
>
>
>
> On 8/12/19, Fred Lunnon <fred.lunnon at gmail.com> wrote:
> > T(n,n)  =  n!  =  n T(n-1,n-1) ;
> >
> > T(n,n-1)  =  (n+1)T(n-1,n-2) - T(n-1,n-1)  =  n! (n+3)/2 ;
> >
> > T(n,n-2)  =  ?
> >
> > T(n,0)  =  2 T(n-1,0) + T(n-2,0) ;  fails for  n = 2  ?!
> >
> > T(n,1)  =  ?
> >
> >
> > On 8/12/19, Neil Sloane <njasloane at gmail.com> wrote:
> >> Dear SeqFans, I have been cleaning up some old sequences with keyword
> >> "uned".
> >> At the end of the weekend, this one is still a mystery: A118980.  The
> >> puzzle boils down to this:
> >> What is the triangle?
> >>
> >> 1;
> >>
> >> 2, 1;
> >>
> >> 6, 5, 2;
> >>
> >> 14, 22, 18, 6;
> >>
> >> 34, 85, 118, 84, 24;
> >>
> >> 82, 311, 660, 780, 480, 120;
> >>
> >> ...
> >>
> >>
> >>
> >> *All that is needed is a plausible guess for a way to continue this
> >> triangle.*
> >>
> >>
> >> The left border is said to be the "companion Pell sequence A099425", and
> >> the right border is the factorials.
> >>
> >> There is an explicit definition in terms of a missing sequence referred
> >> to
> >> as "A118979" (different from the current A118979).
> >>
> >> There are more clues involving A118981, which also need editing,
> (Richard
> >> Mathar suggested a way to edit it in 2011, but I have not touched it
> >> because I did not want to disturb the crime scene).
> >>
> >> A118980 is a "cold case".  Richard Mathar worked on it in 2011, and I
> >> struggled with it over the weekend.  Now I'm calling in "The Wolf" - the
> >> combined minds of the Sequence Fans list. (The reference is to Pulp
> >> Fiction.)
> >>
> >> --
> >> Seqfan Mailing list - http://list.seqfan.eu/
> >>
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

```