A320666 a(n) is the maximum number of liberties a single group can have on an otherwise empty n X n Go board.

0, 2, 6, 9, 14, 22, 29, 38, 51, 61, 74, 92, 105, 122, 145, 161, 182, 210, 229, 254, 287, 309, 338, 376

Offset

1,2

Comments

For 1 X 1 the solution is a single stone on the only possible position and is not a valid final board state in a real game of Go.

Also seems to be the answer to the following parking problem: maximum number of cars in an n X n carpark such that any car can leave through a single exit. See Puzzling StackExchange links. - Dmitry Kamenetsky, Mar 26 2021

Links

Table of n, a(n) for n=1..24.

Ton Hospel, Table of n, a(n) for n = 1..24

Puzzling StackExchange, Placing 9 cars into a 4x4 carpark, March 2021.

Puzzling StackExchange, A special parking lot, October 2017.

Formula

Exact for n <= 24, conjectured for n > 24 but it is at least a lower bound:

a(n) = 0 if n = 1.

a(n) = 2 if n = 2.

a(n) = 6 if n = 3.

a(n) = n*(2*n-1)/3 if n = 0 (mod 3) and n != 3.

a(n) = ((2n-1)^2+5)/6 if n = 1 (mod 3) and n != 1.

a(n) = ((2n-1)^2+3)/6 if n = 2 (mod 3).

Conjectures from Colin Barker, Jun 05 2019: (Start)

G.f.: x^2*(2 + 4*x + 3*x^2 + x^3 + x^5 + x^6 + x^7 - x^8) / ((1 - x)^3*(1 + x + x^2)^2).

a(n) = a(n-1) + 2*a(n-3) - 2*a(n-4) - a(n-6) + a(n-7) for n>9.

(End)

Example

For n = 7 one of many a(7) = 29 solutions:
  *********
  *.O.....*
  *.OOOOOO*
  *.O....O*
  *.O.....*
  *.O.OOO.*
  *.OOO.O.*
  *.O...O.*
  *********

Prog

(Perl)
sub a {
     # Conjectured: This program is valid for any m X n board size
     my ($m, $n) = @_;
     $n = $m if !defined $n;
     ($m, $n) = ($n, $m) if $m > $n;
     # So now $m <= $n
     # This program is certain to be valid for all $m <= 24
     if ($m >= 4) {
         return $m*(2*$n-1)/3 if $m % 3 == 0;
         return $n*(2*$m-1)/3 if $n % 3 == 0;
         return ((2*$m-1)*(2*$n-1)+5)/6 if $m % 3 == 1 && $n % 3 == 1;
         return ((2*$m-1)*(2*$n-1)+3)/6; # if $m % 3 == 2 || $n % 3 == 2
     }
     return 2*$n if $m == 3;
     return $n == 3 ? 4 : $n if $m == 2;
     return $n >= 3 ? 2 : $n-1 if $m == 1;
     die "Bad call";
}

Crossrefs

A071619 is a trivial upper bound for this sequence.

Sequence in context: A096378 A342426 A217001 * A079023 A327967 A189760

Adjacent sequences: A320663 A320664 A320665 * A320667 A320668 A320669

Keyword

nonn,more

Author

Ton Hospel, Oct 28 2018

Status

approved