

A320666


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


1



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


LINKS

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


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*n1)/3 if n = 0 (mod 3) and n != 3.
a(n) = ((2n1)^2+5)/6 if n = 1 (mod 3) and n != 1.
a(n) = ((2n1)^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(n1) + 2*a(n3)  2*a(n4)  a(n6) + a(n7) 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 $m*(2*$n1)/3 if $m % 3 == 0;
return $n*(2*$m1)/3 if $n % 3 == 0;
return ((2*$m1)*(2*$n1)+5)/6 if $m % 3 == 1 && $n % 3 == 1;
return ((2*$m1)*(2*$n1)+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 : $n1 if $m == 1;
die "Bad call";
}


CROSSREFS

A071619 is a trivial upper bound for this sequence.
Sequence in context: A327895 A096378 A217001 * A079023 A327967 A189760
Adjacent sequences: A320663 A320664 A320665 * A320667 A320668 A320669


KEYWORD

nonn,more


AUTHOR

Ton Hospel, Oct 28 2018


STATUS

approved



