Sequence A112088 without example

[Max]

On 6/6/06, Hugo Pfoertner <all at abouthugo.de> wrote:

> AFAIK, Rainer is talking about the Josephus Problem in its classical
> form, where n people are arranged around the circumference of a circle.
> An executioner walks around the circle without changing his direction
> and without making shortcuts. Starting at the first person, every third
> (of the remaining) persons on the circumference of the circle is
> eliminated until only one person remains. The surviving persons remain
> at their original positions in the circle. Therefore the executioner
> needs to walk a certain amount of rounds until his job is done. Rainers
> observation is that A112088(n) is the minimum number of people required
> such that the executioner needs n rounds until only one survivor
> remains.

Ops! I've tried to recompute that sequence myself and I've got
sequence A005428(n+1) which is different from Rainer's one.

In particular, for n=7 we have A005428(7+1)=21 while Rainer's value is 24.
Let see how the executioner proceeds at 21 people.
At the beginning we have
0: * 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
where * is the executioner's position.
After first round we left with
1: * 1 2 4 5 7 8 10 11 13 14 16 17 19 20
after second:
2: * 2 5 7 10 11 14 16 19 20
after third:
3: * 2 5 10 11 16 19
after fourth:
4: * 2 5 11 16
after fifth:
5: 2 * 16
the next step of the executioner makes two rounds (sixth and seventh)
at a time ending with
7: * 2

So 21 people require 7 rounds to be eliminated, then according to
Hugo's definition the the 7th element of the discussed sequence must
be <=21.

What's wrong?

Thanks,
Max

> > On 6/5/06, Rainer Rosenthal <r.rosenthal at web.de> wrote:
> > > To SeqFan and to author Simon Strandgaard:
> > >
> > > Playing the "Josephus problem" with "every third out"
> > > and denoting the minimum numbers in this game, I found
> > >
> > >      2, 3, 5, 7, 11, 16, 24, 36, 54, 81, ...
> > >
> > > I was happy to find them in this sequence:
> > > http://www.research.att.com/~njas/sequences/A112088.
> > >
> > > My computed values are the same up to a(18) = 2082, so
> > > that my Josephus game is likely to generate exactly
> > > this sequence.
> [...]
>