A202657 -id:A202657 - cp's OEIS frontend

A202654 Number of ways to place 3 nonattacking semi-queens on an n X n board.

0, 0, 3, 52, 370, 1620, 5285, 14168, 33012, 69240, 133815, 242220, 415558, 681772, 1076985, 1646960, 2448680, 3552048, 5041707, 7018980, 9603930, 12937540, 17184013, 22533192, 29203100, 37442600, 47534175, 59796828, 74589102, 92312220, 113413345, 138388960

Offset: 1

Vaclav Kotesovec, Dec 22 2011

nonn

Two semi-queens do not attack each other if they are in the same northwest-southeast diagonal.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Christopher R. H. Hanusa, Thomas Zaslavsky, A q-queens problem. VII. Combinatorial types of nonattacking chess riders, arXiv:1906.08981 [math.CO], 2019.
V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A099152, A047659, A103220, A202655, A202656, A202657.

Rest@ CoefficientList[Series[-x^3*(17 x^3 + 69 x^2 + 31 x + 3)/(x - 1)^7, {x, 0, 32}], x] (* Michael De Vlieger, Aug 19 2019 *)

a(n) = 1/6*(n-2)*(n-1)*n*(n^3-5*n^2+8*n-3).

G.f.: -x^3*(17*x^3 + 69*x^2 + 31*x + 3)/(x-1)^7.

A202655 Number of ways to place 4 nonattacking semi-queens on an n X n board.

Original entry on oeis.org

0, 0, 0, 7, 223, 2429, 15045, 66122, 230074, 675798, 1745318, 4073993, 8764753, 17630795, 33522531, 60756612, 105666148, 177293340, 288246972, 455749371, 702898611, 1060173961, 1567213681, 2274896558, 3247759614, 4566786770, 6332604226, 8669120733, 11727651845

Offset: 1

Vaclav Kotesovec, Dec 22 2011

nonn

Two semi-queens do not attack each other if they are in the same northwest-southeast diagonal.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Christopher R. H. Hanusa, Thomas Zaslavsky, A q-queens problem. VII. Combinatorial types of nonattacking chess riders, arXiv:1906.08981 [math.CO], 2019.
V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (7,-19,21,6,-42,42,-6,-21,19,-7,1).

Cf. A099152, A061994, A103220, A202654, A202656, A202657.

Rest@ CoefficientList[Series[-x^4*(151 x^6 + 1022 x^5 + 2233 x^4 + 2132 x^3 + 1001 x^2 + 174 x + 7)/((x - 1)^9*(x + 1)^2), {x, 0, 29}], x] (* Michael De Vlieger, Aug 19 2019 *)
LinearRecurrence[{7,-19,21,6,-42,42,-6,-21,19,-7,1},{0,0,0,7,223,2429,15045,66122,230074,675798,1745318},30] (* Harvey P. Dale, Mar 17 2025 *)

a(n) = n^8/24 - 2*n^7/3 + 41*n^6/9 - 257*n^5/15 + 341*n^4/9 - 97*n^3/2 + 2341*n^2/72 - 87*n/10 + (n/2 - 1/2)*floor(n/2).

G.f.: -x^4*(151*x^6 + 1022*x^5 + 2233*x^4 + 2132*x^3 + 1001*x^2 + 174*x + 7)/((x-1)^9*(x+1)^2).

A202656 Number of ways to place 5 nonattacking semi-queens on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 23, 1104, 16945, 141696, 810746, 3568352, 12948318, 40514560, 112720393, 285073712, 666143975, 1456288512, 3007576740, 5913372864, 11138305068, 20202100224, 35433809451, 60316600080, 99947225741, 161638967424, 255701773822, 396439174560, 603407582570

Offset: 1

Vaclav Kotesovec, Dec 22 2011

nonn

Two semi-queens do not attack each other if they are in the same northwest-southeast diagonal.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Christopher R. H. Hanusa, Thomas Zaslavsky, A q-queens problem. VII. Combinatorial types of nonattacking chess riders, arXiv:1906.08981 [math.CO], 2019.
V. Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (7, -17, 7, 43, -77, 11, 99, -99, -11, 77, -43, -7, 17, -7, 1).

Cf. A099152, A108792, A103220, A202654, A202655, A202657.

Rest@ CoefficientList[Series[-x^5*(1899 x^9 + 16515 x^8 + 60512 x^7 + 116784 x^6 + 137646 x^5 + 98222 x^4 + 41688 x^3 + 9608 x^2 + 943 x + 23)/((x - 1)^11*(x + 1)^4), {x, 0, 27}], x] (* Michael De Vlieger, Aug 19 2019 *)

a(n) = n^10/120 - 2*n^9/9 + 95*n^8/36 - 183*n^7/10 + 14663*n^6/180 - 1201*n^5/5 + 16753*n^4/36 - 25364*n^3/45 + 68293*n^2/180 - 12781*n/120 + (n^3/2 - 6*n^2 + 39*n/2 - 61/4)*floor(n/2).

G.f.: -x^5*(1899*x^9 + 16515*x^8 + 60512*x^7 + 116784*x^6 + 137646*x^5 + 98222*x^4 + 41688*x^3 + 9608*x^2 + 943*x + 23)/((x-1)^11*(x+1)^4).

A342372 Triangle T(n,k) of number of ways of arranging q nonattacking semi-queens on an n X n toroidal board, where 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 0, 1, 9, 9, 3, 1, 16, 48, 32, 0, 1, 25, 150, 250, 75, 15, 1, 36, 360, 1200, 1224, 288, 0, 1, 49, 735, 4165, 8869, 6321, 931, 133, 1, 64, 1344, 11648, 43136, 64512, 33024, 4096, 0, 1, 81, 2268, 27972, 160866, 423306, 469800

Offset: 1

Walter Trump, Mar 09 2021

T(0,0):=1 for combinatorial reasons.
A semi-queen can only move horizontal, vertical and parallel to the main diagonal of the board. Moves parallel to the secondary diagonal are not allowed.
Instead of a board on a torus, you can imagine that the semi-queens can leave a flat board on one side and re-enter the board on the other side.

			  1;
  1,  1;
  1,  4,   0;
  1,  9,   9,   3;
  1, 16,  48,  32,  0;
  1, 25, 150, 250, 75, 15;

Walter Trump, Table of n, a(n) for n = 1..222
Walter Trump, Semi-queen problem

Cf. A006717, A099152, A103220, A202654, A202655, A202656, A202657.

T(n,0) = 1.
T(n,1) = n^2.
T(n,2) = n^2*(n-1)*(n-2)/2.
T(n,3) = n^2*(n-1)*(n-2)*(n^2-6n+10)/6.
T(2n+1,2n+1) = A006717(n).
T(2n,2n) = 0.

cp's OEIS Frontend

A202654 Number of ways to place 3 nonattacking semi-queens on an n X n board.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A202655 Number of ways to place 4 nonattacking semi-queens on an n X n board.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A202656 Number of ways to place 5 nonattacking semi-queens on an n X n board.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A342372 Triangle T(n,k) of number of ways of arranging q nonattacking semi-queens on an n X n toroidal board, where 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula