# [seqfan] Re: Filling a triangle with numbers

Olivier Gerard olivier.gerard at gmail.com
Fri Jul 6 17:12:41 CEST 2012

```Dear Neil,

I evidently confirm Ron's numbers.  A003121 is what you were looking for.

If one does not require increasing rows, (i.e. the top number is between
[min() , max()]
of the two numbers below the sequence starts

1, 2, 20, 1744, ...   (apparently not in the OEIS)

If one does not require that a number be used only one time, but with strict
inequality, and increasing rows, the sequence starts

1, 1, 8, ...

If one does not require increasing rows or unique use, the sequence starts

1, 2, 104, ...

Olivier

On Fri, Jul 6, 2012 at 4:31 PM, Ron Hardin <rhhardin at att.net> wrote:

> sequence starts   1  1  2  12  286
> which matches A003121
>
>  rhhardin at mindspring.com
> rhhardin at att.net (either)
>
>
>
> ----- Original Message ----
> > From: Ron Hardin <rhhardin at att.net>
> > To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> > Sent: Fri, July 6, 2012 10:18:38 AM
> > Subject: [seqfan] Re: Filling a triangle with numbers
> >
> > I get 12 for  a(4)
> > .....5........6........6........5........5........3........3........4..
> > ....3.7......3.7......2.7......2.7......3.6......2.6......2.7......3.7..
> > ...1.4.8....1.4.8....1.4.8....1.4.8....1.4.8....1.5.8....1.5.8....1.5.8..
> >
> ..0.2.6.9..0.2.5.9..0.3.5.9..0.3.6.9..0.2.7.9..0.4.7.9..0.4.6.9..0.2.6.9..
> > ..
> > .....4........4........5........4..
> > ....2.6......2.7......2.6......3.6..
> > ...1.5.8....1.5.8....1.4.8....1.5.8..
> > ..0.3.7.9..0.3.6.9..0.3.7.9..0.2.7.9..
> >
> > rhhardin at mindspring.com
> > rhhardin at att.net (either)
> >
> >
> >
> > ----- Original Message ----
> > > From: Neil Sloane  <njasloane at gmail.com>
> > > To:  Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> > > Sent:  Fri, July 6, 2012 9:51:14 AM
> > > Subject: [seqfan] Filling a triangle with  numbers
> > >
> > > I heard this problem in Hungary recently:
> > >
> > > How many ways are there to  arrange the numbers 1 through n(n+1)/2  in
> > > a triangle so that each term is  between the two numbers
> > >  immediately below it (and each row is  increasing)?
> > >
> > >  Examples:
> > > .2.
> > > 1.3
> > >
> > > ..4..
> > > .2.5.
> > >  1.3.6
> > >
> > > ..3..
> > > .2.5.
> > > 1.4.6
> > >
> > >  I  get a(1)=a(2)=1, a(3)=2, a(4)=6?
> > >
> > > The only related  sequences I can find  in the OEIS are A064049,
> A064059,
> > > which are  different. However, it is quite  likely this is a
> well-studied
> > >  problem.
> > >
> > > Also, what if one omits the  clause in  parentheses in the  definition?
> > >
> > > Neil
> > >
> > >  _______________________________________________
> > >
> > > Seqfan   Mailing list - http://list.seqfan.eu/
> > >
> >
> > _______________________________________________
> >
> > Seqfan Mailing  list - http://list.seqfan.eu/
> >
>
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>
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>

```