A085801 -id:A085801 - cp's OEIS frontend

A201629 a(n) = n if n is even and otherwise its nearest multiple of 4.

0, 0, 2, 4, 4, 4, 6, 8, 8, 8, 10, 12, 12, 12, 14, 16, 16, 16, 18, 20, 20, 20, 22, 24, 24, 24, 26, 28, 28, 28, 30, 32, 32, 32, 34, 36, 36, 36, 38, 40, 40, 40, 42, 44, 44, 44, 46, 48, 48, 48, 50, 52, 52, 52, 54, 56, 56, 56, 58, 60, 60, 60, 62, 64, 64, 64, 66, 68, 68

Offset: 0

Vaclav Kotesovec, Dec 03 2011

For n > 1, the maximal number of nonattacking knights on a 2 x (n-1) chessboard.
Compare this with the binary triangle construction of A240828.
Minimal number of straight segments in a rook circuit of an (n-1) X n board (see example). - Ruediger Jehn, Feb 26 2021

			G.f. = 2*x^2 + 4*x^3 + 4*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 8*x^8 + 8*x^9 + ...
From _Ruediger Jehn_, Feb 26 2021: (Start)
a(5) = 4:
   +----+----+----+----+----+
   |  __|____|_   |   _|__  |
   | /  |    | \  |  / |  \ |
   +----+----+----+----+----+
   | \__|__  | |  |  | |  | |
   |    |  \ | \__|__/ |  | |
   +----+----+----+----+----+
   |  __|__/ |  __|__  |  | |
   | /  |    | /  |  \ |  | |
   +----+----+----+----+----+
   | \  |    | |  |  | |  | |
   |  \_|____|_/  |  \_|__/ |
   +----+----+----+----+----+
There are at least 4 squares on the 4 X 5 board with straight lines (here in squares a_12, a_25, a_35 and a_42).  (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Rüdiger Jehn, Properties of Hamiltonian Circuits in Rectangular Grids, arXiv:2103.15778 [math.GM], 2021.
Craig Knecht, Row sums of superimposed and added binary filled triangles.
Vaclav Kotesovec, Non-attacking chess pieces

Cf. A004524, A085801, A189889, A190394.

a201629 = (* 2) . a004524 . (+ 1) -- Reinhard Zumkeller, Aug 05 2014

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(2*x^2/((1-x)^2*(1+x^2)))); // G. C. Greubel, Aug 13 2018

seq(n-sin(Pi*n/2), n=0..30); # Robert Israel, Jul 14 2015

Table[2*(Floor[(Floor[(n + 1)/2] + 1)/2] + Floor[(Floor[n/2] + 1)/2]), {n, 1, 100}]
Table[If[EvenQ[n], n, 4*Round[n/4]], {n, 0, 68}] (* Alonso del Arte, Jan 27 2012 *)
CoefficientList[Series[2 x^2/((-1 + x)^2 (1 + x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 06 2014 *)
a[ n_] := n - KroneckerSymbol[ -4, n]; (* Michael Somos, Jul 18 2015 *)

a(n)=n\4*4+[0, 0, 2, 4][n%4+1] \\ Charles R Greathouse IV, Jan 27 2012

{a(n) = n - kronecker( -4, n)}; /* Michael Somos, Jul 18 2015 */

a(n) = n - sin(n*Pi/2).
G.f.: 2*x^2/((1-x)^2*(1+x^2)).
a(n) = 2*A004524(n+1). - R. J. Mathar, Feb 02 2012
a(n) = n+(1-(-1)^n)*(-1)^((n+1)/2)/2. - Bruno Berselli, Aug 06 2014
E.g.f.: x*exp(x) - sin(x). - G. C. Greubel, Aug 13 2018

Formula corrected by Robert Israel, Jul 14 2015

A189889 Maximum number of nonattacking kings on an n X n toroidal board.

Original entry on oeis.org

1, 1, 1, 4, 5, 9, 10, 16, 18, 25, 27, 36, 39, 49, 52, 64, 68, 81, 85, 100, 105, 121, 126, 144, 150, 169, 175, 196, 203, 225, 232, 256, 264, 289, 297, 324, 333, 361, 370, 400, 410, 441, 451, 484, 495, 529, 540, 576, 588, 625

Offset: 1

Vaclav Kotesovec, Apr 30 2011

a(n) is the independence number of the Cayley graph on the group Z_n X Z_n with generators (+-e_1, +-e_2)<>(0,0) where e_i is in {0,1} for i=1,2. - Miquel A. Fiol, Aug 07 2024

For n>=4 a(n) is the maximum number of edges of an n-cycle graph with chords not containing any triangle with some edges of the cycle. - Miquel A. Fiol, Sep 20 2024

John Watkins, Across the Board: The Mathematics of Chessboard Problems (2004), Theorem 11.1, p.194.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Hernan de Alba, W. Carballosa, J. Leaños, and L. M. Rivera, Independence and matching numbers of some token graphs, arXiv preprint arXiv:1606.06370 [math.CO], 2016.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 751.
Eric Weisstein's World of Mathematics, Kings Problem.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,-1,1).

Cf. A018807, A085801, A172158, A174558, A179428, A180067.

[1] cat [Floor(n*Floor(n/2)/2): n in [2..50]]; // G. C. Greubel, Jan 13 2018

A189889:=n->`if`(n=1,1,floor(n*floor(n/2)/2)); seq(A189889(k), k=1..100); # Wesley Ivan Hurt, Nov 07 2013

Table[If[n==1,1,Floor[(n*Floor[n/2])/2]],{n,1,50}]
CoefficientList[Series[(- x^7 + x^6 + x^5 + 3 * x^3 - x^2 + 1) / (-x^7 + x^6 + x^5 - x^4 + x^3 - x^2 - x + 1), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 02 2013 *)
Join[{1},LinearRecurrence[{1,1,-1,1,-1,-1,1},{1,1,4,5,9,10,16},50]] (* Harvey P. Dale, Aug 07 2013 *)

Vec(x*(-x^7 + x^6 + x^5 + 3*x^3 - x^2 + 1) / (-x^7 + x^6 + x^5 - x^4 + x^3 - x^2 - x + 1) + O(x^51)) \\ Indranil Ghosh, Mar 09 2017

a(n) = if(n==1, 1, floor((n*floor(n/2))/2)); \\ Indranil Ghosh, Mar 09 2017

def A189889(n): return 1 if n==1 else (n*(n/2))/2 # Indranil Ghosh, Mar 09 2017

a(n) = floor((n*floor(n/2))/2), n > 1 (Watkins and Ricci, 2004).
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7).
G.f.: x*(-x^7 +x^6 +x^5 +3*x^3 -x^2 +1) / (-x^7 +x^6 +x^5 -x^4+ x^3 -x^2 -x +1).

A279402 Domination number for queen graph on an n X n toroidal board.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 4, 4, 5, 5, 5, 6, 7, 7, 5, 8, 9, 8, 10, 10, 7, 11

Offset: 1

Andrey Zabolotskiy, Dec 11 2016

That is, the minimal number of queens needed to cover an n X n toroidal chessboard so that every square either has a queen on it, or is under attack by a queen, or both.
Row lengths of the triangle A279403.
All dominating sets are translation-invariant on the torus.
a(4*n) <= 2*n.
a(n) <= A075458(n).

			The minimal dominating set for the queens' graph on a 15 X 15 toroidal board is:
...............
..........Q....
...............
...............
.Q.............
...............
...............
.......Q.......
...............
...............
.............Q.
...............
...............
....Q..........
...............
Hence a(15) = 5.

John J. Watkins, Across the Board: The Mathematics of Chessboard Problem, Princeton University Press, 2004, pp. 139-140.

A. P. Burger and C. M. Mynhardt, The domination number of the toroidal queens graph of size 3k × 3k, Australasian Journal of Combinatorics, 28 (2003), 137-148.
Andy Huchala, Python program.
Christina M. Mynhardt, Upper bounds for the domination numbers of toroidal queens graphs, Discussiones Mathematicae Graph Theory, 23 (2003), 163-175.

Cf. A075458, A274138, A279403, A279404, A279405, A279406, A279407, A279408, A279409.

a(3*n) = n if n == 1, 5, 7, 11 (mod 12);
a(3*n) = n+1 if n == 2, 10 (mod 12);
a(3*n) = n+2 otherwise.
I.e., a(3*n) = 2*n - A085801(n).

a(16)-a(22) from Andy Huchala, Mar 04 2024

A279404 Independent domination number for queens' graph on an n X n toroidal board.

Original entry on oeis.org

1, 1, 1, 2, 5, 4, 5, 4, 5, 5, 5, 6, 7, 7, 5, 8, 9, 10

Offset: 1

Andrey Zabolotskiy, Dec 11 2016

That is, the minimal number of queens needed to cover an n X n toroidal chessboard so that every square either has a queen on it or is under attack by a queen, but not both.

A279402(n) <= a(n) <= A085801(n).

Andy Huchala, Python program.
Christina M. Mynhardt, Upper bounds for the domination numbers of toroidal queens graphs, Discussiones Mathematicae Graph Theory, 23 (2003), 163-175.

Cf. A075324, A085801, A279402.

a(3*n) = n if n = 1, 5, 7, 11 (mod 12).

a(17)-a(18) from Andy Huchala, Mar 09 2024

A190394 Maximum number of nonattacking nightriders on an n X n board.

Original entry on oeis.org

1, 4, 5, 8, 10, 16, 17, 20, 21, 24, 26, 32, 33, 36, 39, 42, 45, 48, 51, 54, 58, 64, 65, 66, 68, 72, 75, 80, 81, 84, 87, 90, 93

Offset: 1

Vaclav Kotesovec, May 10 2011

A nightrider is a fairy chess piece that can move any distance in a direction specified by a knight move.

Maximum number of nonattacking nightriders on an n X n toroidal board is n.

			From _Rob Pratt_, Jul 24 2015: (Start)
a(20) = 54:
  XX--XXXX---X------XX
  XX---------X--XX--XX
  --------------------
  ---X----------------
  X-----------------X-
  X-----------------X-
  X-------------------
  X---------X---------
  ------------------XX
  ------------X-------
  -------X------------
  XX------------------
  ---------X---------X
  -------------------X
  -X-----------------X
  -X-----------------X
  ----------------X---
  --------------------
  XX--XX--X---------XX
  XX------X---XXXX--XX
(End)

Andy Huchala, Python program.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, pp. 751-763.
Rob Pratt, 54 nonattacking nightriders on a 20 X 20 board.

Cf. A085801, A190393, A172141, A173429.

2n <= a(n) <= 3n-2, for n > 3.

a(n) >= 24*floor((n+4)/10)-8, for n >= 6. - Vaclav Kotesovec, Apr 01 2012

Terms a(11)-a(16) from Vaclav Kotesovec, May 13 2011
Terms a(17)-a(19) from Vaclav Kotesovec, Apr 01 2012
a(20) from Rob Pratt, Jul 24 2015
a(21)-a(32) from Paul Tabatabai, Nov 06 2018
a(33) from Andy Huchala, Mar 30 2024

A279409 Triangle read by rows: T(n,m) (n>=m>=1) = maximum number of nonattacking kings on an n X m toroidal board.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 4, 2, 2, 2, 4, 5, 3, 3, 3, 6, 6, 9, 3, 3, 3, 6, 7, 9, 10, 4, 4, 4, 8, 8, 12, 12, 16, 4, 4, 4, 8, 9, 12, 13, 16, 18, 5, 5, 5, 10, 10, 15, 15, 20, 20, 25, 5, 5, 5, 10, 11, 15, 16, 20, 22, 25, 27, 6, 6, 6, 12, 12, 18, 18, 24, 24, 30, 30, 36

Offset: 1

Andrey Zabolotskiy, Dec 16 2016

Independence number of the kings' graph on toroidal n X m chessboard.
Right border T(n,n) is A189889.
For the usual non-toroidal case, the formula is ceiling(m/2)*ceiling(n/2).

			Triangle starts:
  1;
  1, 1;
  1, 1, 1;
  2, 2, 2, 4;
  2, 2, 2, 4, 5;
  3, 3, 3, 6, 6, 9;
  3, 3, 3, 6, 7, 9, 10;
  ...

John J. Watkins, Across the Board: The Mathematics of Chessboard Problem, Princeton University Press, 2004, pages 194-196.

Indranil Ghosh, Rows 1..125, flattened
Dan Freeman, Chessboard Puzzles Part 4 - Other Surfaces and Variations.
Vaclav Kotesovec, Non-attacking chess pieces.

Cf. A085801, A189889, A279408.

T[1, 1] = 1; T[n_, m_]:= If[m==1, Floor[n/2], Floor[Min[m Floor[n/2], n Floor[m/2]]/2]]; Flatten[Table[T[n,m], {n, 1, 12},{m, 1,n}]] (* Indranil Ghosh, Mar 09 2017 *)

tabl(nn) = {for(n=1, 12, for(m=1, n, print1(if(m==1,if(n==1, 1, floor(n/2)), floor(min(m*floor(n/2), n*floor(m/2))/2)),", ");); print();); };
tabl(12); \\ Indranil Ghosh, Mar 09 2017

def T(n,m):
    if m==1:
        if n==1: return 1
        return n//2
    return min(m*(n//2), n*(m//2))//2
i=1
for n in range(1,126):
    for m in range(1, n+1):
        print(i, T(n,m))
        i+=1 # Indranil Ghosh, Mar 09 2017

T(n,m) = floor(min(m*floor(n/2), n*floor(m/2))/2) for m>1;

T(n,1) = floor(n/2) for n>1.

A133143 Maximal number of mutually nonattacking Super Queens on an n X n board. (A Super Queen is a queen with both queen and knight powers.)

Original entry on oeis.org

1, 1, 1, 2, 4, 4, 5, 6, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

Offset: 1

Rachit Agrawal (rachit_agrawal(AT)daiict.ac.in), Dec 16 2007

nonn

From Vaclav Kotesovec, Mar 14 2011: (Start)
For n >= 10, a(n)=n, see A051223.
For same problem on a toroidal chessboard the results for n > 10 are the same as for queens (A085801). (End)

			a(4) = 2:
|X|0|0|0|
|0|0|0|X|
|0|0|0|0|
|0|0|0|0|
where X denotes the place of a Super Queen.

Bill Butler, More Information
Rachit Agrawal, Java code for this sequence

// See Java code text file in the links section.

A178986 Number of ways to place n nonattacking amazons (superqueens) on an n X n toroidal board.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 44, 0, 1092, 0, 0, 0, 16932, 0, 24776, 0, 0, 0, 1881492, 0

Offset: 1

Vaclav Kotesovec, Jan 03 2011

An amazon (superqueen) moves like a queen and a knight.

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013.
Paul Monsky, Superqueens: Problem E3162, American Mathematical Monthly, vol. 96 (1989), p.258-259.

Cf. A051223, A051906, A085801, A000170, A178975.

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A201629 a(n) = n if n is even and otherwise its nearest multiple of 4.

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A189889 Maximum number of nonattacking kings on an n X n toroidal board.

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A279402 Domination number for queen graph on an n X n toroidal board.

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A279404 Independent domination number for queens' graph on an n X n toroidal board.

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A190394 Maximum number of nonattacking nightriders on an n X n board.

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A279409 Triangle read by rows: T(n,m) (n>=m>=1) = maximum number of nonattacking kings on an n X m toroidal board.

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A133143 Maximal number of mutually nonattacking Super Queens on an n X n board. (A Super Queen is a queen with both queen and knight powers.)

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