A071619 -id:A071619 - cp's OEIS frontend

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

Ton Hospel, Oct 28 2018

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

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

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.

A071619 is a trivial upper bound for this sequence.

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";
}

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)

A358160 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = ij - floor(ij/3).

Original entry on oeis.org

1, 2, 40, 3884, 1016376, 534983256, 510252517152, 802452895865280, 1901953775079849600, 6537796866589765507200, 31381746234057256630521600

Offset: 0

Stefano Spezia, Nov 01 2022

The matrix M(n) is the n-th principal submatrix of the rectangular array A143976.

			a(2) = 40:
    1   2   2   3
    2   3   4   6
    2   4   6   8
    3   6   8  11

Wikipedia, Hafnian
Wikipedia, Symmetric matrix

Cf. A143976.

Cf. A071619 (matrix element M[n,n]), A358159 (permanent of M(2*n)), A358042 (trace of M(n)).

M[i_, j_, n_]:=Part[Part[Table[r*c-Floor[r*c/3], {r, n}, {c, n}], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]

tm(n) = matrix(n, n, i, j, i*j - (i*j)\3);
a(n) = my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]); ); s/(n!*2^n); \\ Michel Marcus, May 02 2023

a(6) from Michel Marcus, May 02 2023

a(7)-a(10) from Pontus von Brömssen, Oct 15 2023

A357837 a(n) is the sum of the lengths of all the segments used to draw a square of side n representing a fishbone pattern using symmetric L-shaped tiles with side length 2.

Original entry on oeis.org

0, 4, 10, 20, 32, 46, 64, 84, 106, 132, 160, 190, 224, 260, 298, 340, 384, 430, 480, 532, 586, 644, 704, 766, 832, 900, 970, 1044, 1120, 1198, 1280, 1364, 1450, 1540, 1632, 1726, 1824, 1924, 2026, 2132, 2240, 2350, 2464, 2580, 2698, 2820, 2944, 3070, 3200, 3332

Offset: 0

Stefano Spezia, Oct 17 2022

			Illustrations for n = 1..8:
        _           _ _          _ _ _
       |_|         |  _|        |  _|_|
                   |_|_|        |_|  _|
                                |_|_|_|
    a(1) = 4     a(2) = 10     a(3) = 20
     _ _ _ _     _ _ _ _ _    _ _ _ _ _ _
    |  _|_| |   |  _|_|  _|  |  _|_|  _|_|
    |_|  _|_|   |_|  _|_| |  |_|  _|_|  _|
    |_|_|  _|   |_|_|  _|_|  |_|_|  _|_| |
    |_ _|_|_|   |  _|_|  _|  |  _|_|  _|_|
                |_|_ _|_|_|  |_|  _|_|  _|
                             |_|_|_ _|_|_|
    a(4) = 32    a(5) = 46     a(6) = 64
      _ _ _ _ _ _ _      _ _ _ _ _ _ _ _
     |  _|_|  _|_| |    |  _|_|  _|_|  _|
     |_|  _|_|  _|_|    |_|  _|_|  _|_| |
     |_|_|  _|_|  _|    |_|_|  _|_|  _|_|
     |  _|_|  _|_| |    |  _|_|  _|_|  _|
     |_|  _|_|  _|_|    |_|  _|_|  _|_| |
     |_|_|  _|_|  _|    |_|_|  _|_|  _|_|
     |_ _|_|_ _|_|_|    |  _|_|  _|_|  _|
                        |_|_ _|_|_ _|_|_|
        a(7) = 84           a(8) = 106

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A002264, A002522, A005843, A047410 (first differences), A071619, A211547.

Cf. A345118.

Table[2(Ceiling[2(n+1)^2/3]-1),{n,0,49}]

a(n) = 2*(ceiling(2*(n+1)^2/3) - 1).
a(n) = 2*(A071619(n+1) - 1).
a(n) = 2*(1 + n^2 - 2*(n - 2)*floor((n - 1)/3) + 3*floor((n - 1)/3)^2) for n > 0.
a(n) = Sum_{k=1..n} A047410(k+1) for n > 0.
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n > 4.
O.g.f.: 2*x*(2 + x + 2*x^2 - x^3)/((1 - x)^3*(1 + x + x^2)).
E.g.f.: 2*exp(-x/2)*(exp(3*x/2)*(6*x*(3 + x) - 1) + cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/9.

cp's OEIS Frontend

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

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A358160 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = ij - floor(ij/3).

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A357837 a(n) is the sum of the lengths of all the segments used to draw a square of side n representing a fishbone pattern using symmetric L-shaped tiles with side length 2.

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Perl

Formula

A358160 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = i*j - floor(i*j/3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A357837 a(n) is the sum of the lengths of all the segments used to draw a square of side n representing a fishbone pattern using symmetric L-shaped tiles with side length 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A358160 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = ij - floor(ij/3).