[seqfan] Re: Parking lot for bad drivers

[Vladimir Shevelev]

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,
Vladimir


________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Rob Pratt [Rob.Pratt at sas.com]
Sent: 17 July 2015 22:57
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: Parking lot for bad drivers

XX_XX_XX_XX_X_X
____________X_X
XX_X_X_X_XX____
___X_X_X____X_X
XX_______XX_X_X
___XX_XX_______
XX_______X_X_XX
___X_X_X_X_X___
XX_X_X_______XX
_______XX_XX___
X_X_XX_______XX
X_X____X_X_X___
____XX_X_X_X_XX
X_X____________
X_X_XX_XX_XX_XX

-----Original Message-----
From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Bob Selcoe
Sent: Thursday, July 16, 2015 2:34 AM
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: Parking lot for bad drivers

Hi Rob & Seqfans,

Would you please provide a solution for A260090(15) = 85?  I can only see 84 as maximum.

Thanks,
Bob Selcoe


--------------------------------------------------
From: "Rob Pratt" <Rob.Pratt at sas.com>
Sent: Wednesday, July 15, 2015 3:41 PM
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: Parking lot for bad drivers

> At Neil's request, I created a new entry (will be
> http://oeis.org/A260090).  I described it in terms of attacking kings
> on a chessboard and conjectured a formula.
>
> -----Original Message-----
> From: Rob Pratt
> Sent: Wednesday, July 15, 2015 11:29 AM
> To: Sequence Fanatics Discussion list
> Subject: RE: [seqfan] Re: Parking lot for bad drivers
>
> And here’s a less trivial 26-car solution:
>
> XX_XX_XX
> ________
> XX_XX_XX
> ________
> XX_XX_X_
> _______X
> X_X_X___
> X_X_X_XX
>
> For n in 1,...,8, I get the following, which is not in OEIS:
> 1, 2, 4, 8, 12, 16, 21, 26
>
> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of
> APPLEGATE, DAVID L (DAVID L)
> Sent: Wednesday, July 15, 2015 10:54 AM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: Parking lot for bad drivers
>
> I wonder if I misunderstand the problem as well - as I understand the
> problem, there's a trivial 24-car solution for the 8x8 lot:
>
> XX_XX_XX
> ________
> XX_XX_XX
> ________
> XX_XX_XX
> ________
> XX_XX_XX
> ________
>
> David Applegate
>
> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Rob
> Pratt
> Sent: Wednesday, July 15, 2015 10:45 AM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: Parking lot for bad drivers
>
> Sorry, I just realized that I misinterpreted your problem to mean no
> three in a row, so A181018 provides only an upper bound.
>
> -----Original Message-----
> From: Rob Pratt
> Sent: Wednesday, July 15, 2015 10:42 AM
> To: 'Sequence Fanatics Discussion list'
> Subject: RE: [seqfan] Parking lot for bad drivers
>
> https://oeis.org/A181018
>
> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Dale
> Gerdemann
> Sent: Wednesday, July 15, 2015 6:22 AM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Parking lot for bad drivers
>
> Hello seqfans,
>
> Here's a puzzle from p. 39 of "Let's be Less Stupid" by Patricia Marx:
>
> At the parking lot for bad drivers, you are not allowed to park in a
> spot that is next to two occupied spots. In other words, no three cars
> can be contiguous--vertically, horizontally, or diagonally. ... How
> many cars can the lot fit.
>
> As an example, the book shows an 8x8 "parking lot" (looks like a
> chessboard) and gives a solution fitting 16 cars into this lot.
>
> Is this in the OEIS?
>
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