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 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 (list; graph; refs; listen; history; text; internal format)
 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 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*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: 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

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Last modified December 2 03:22 EST 2020. Contains 338865 sequences. (Running on oeis4.)