# [seqfan] Re: Parking lot for bad drivers

Neil Sloane njasloane at gmail.com
Sat Jul 25 16:34:47 CEST 2015

```Veikko, Yes, please submit your version!

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Email: njasloane at gmail.com

On Sat, Jul 25, 2015 at 4:23 AM, Veikko Pohjola <veikko at nordem.fi> wrote:

> Dear seqfans,
>
> I am considering to submit my version of this parking lot problem (if it
> has not been done yet). The version is what I did call misinterpreted,
> because it refers to an idealized model of the whole: cars which can move
> horizontally to any direction and a parking lot which is an open square
> with permeable borders. It is thus just a mathematical excercise. The task
> is to find for an nxn square the maximum coverage by the occupied
> 1x1-spaces under the constraint that no rows of three occupied spaces can
> appear vertically, horizontally and diagonally.
>
> Actually we have a tiling problem here. The maximum coverage of an nxn
> square can be obtained with a 3x3-size macrotile, but some extra challenge
> is offered by the evidence that a macrotile of one types can solve only 2/3
> of the cases while another, symmetrical, is necessary to solve the rest.
>
> The sequence of the maximum numbers of occupied spaces in an nxn square
> starts as follows: 1, 4, 6, 9, 16, 20, 25, 36, 40, 49, 64, 68, 81, ... (not
> in OEIS).
>
> The original parking lot problem was launched elsewhere and then discussed
> by many other people. Thus, I will proceed to submission of this version
> only if nobody objects.
>
> Regards,
> Veikko
>
>
> Vladimir Shevelev kirjoitti 19.7.2015 kello 12.47:
>
> > On the other hand, if to mean the initial problem on drivers, in case
> n=15
> > we have 102 cars:
> >
> > XX_XX_XX_XX_XX_
> > ______________X
> > XX_XX_XX_XX_X_X
> > XX_XX_XX_XX_XX_
> >
> > XX_XX_XX_XX_X_X
> > XX_XX_XX_XX_XX_
> >
> > XX_XX_XX_XX_X_X
> > XX_XX_XX_XX_X_X
> >
> > XX_XX_XX_XX__XX
> > XX_XX_XX_XX_XX_
> > ______________X
> > XX_XX_XX_XX_X_X
> >
> > I get 1,2,4,8,12,17,24,30,38,48,56,66,80,90,102,...
> > (not in OEIS).
> > Conjecture: 1) if n==1 (mod 3), then a(1)=1,
> > a(n) = 4*(n-1)*(n+2)/9, n>=4;
> > 2) if n==2 (mod 3), then a(n) = 2*(n+1)*(2*n-1)/9, n>=2.
> > 3) if n==0 (mod 3), then a(n) = 4*(n/3)^2+x_n, n>=6. where x_n
> > are small additives which till now I do not know: x_3=0, x_6=1, x_9=2,
> x_12=2, x_15=2.
> >
> > Note that up to n=8, earlier Rob had 1, 2, 4, 8, 12, 16, 21, 26.
> > However, for n=6 we have 17 cars
> >
> > XX_XX_
> > _____X
> > XX_X_X
> > XX_XX
> > ______
> > XX_XX
> >
> > while for n=7 we have 24 cars:
> >
> > XX_XX_X
> > _______
> > XX_XX_X
> > XX_XX_X
> > _______
> > XX_XX_X
> > _X_XX_X
> >
> > and for n=8, as I have already written, we have 30 cars
> > (Veikko's result which was confirmed by me in a simpler variant).
> >
> > Best regards,