A061994 -id:A061994 - cp's OEIS frontend

A172139 Number of ways to place 4 nonattacking zebras on an n X n board.

0, 1, 126, 1168, 7334, 35749, 137970, 438984, 1208246, 2969389, 6662480, 13873100, 27144408, 50389581, 89424014, 152638280, 251834530, 403250693, 628798516, 957543164, 1427453780, 2087456085, 2999819778, 4242915176

Offset: 1

Vaclav Kotesovec, Jan 26 2010

nonn

Zebra is a (fairy chess) leaper [2,3].

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A061994, A172127, A172135, A172137, A172138.

CoefficientList[Series[x(1+117*x+70*x^2+1274*x^3+1333*x^4-2109*x^5-462*x^6 +8858*x^7-17006*x^8+15166*x^9-6838*x^10+1478*x^11-650*x^12+760*x^13-376*x^14 +64*x^15)/(1-x)^9, {x,0,40}], x] (* Vincenzo Librandi, May 27 2013 *)

[0,1,126,1168,7334,35749,137970,438984] + [(n^8 -54*n^6 +240*n^5 +827*n^4 -8592*n^3 +10362*n^2 +75600*n -204864)/24 for n in (9..50)] # G. C. Greubel, Apr 19 2022

a(n) = (n^8 - 54*n^6 + 240*n^5 + 827*n^4 - 8592*n^3 + 10362*n^2 + 75600*n - 204864)/24, n >= 9.

G.f.: x^2*(1 + 117*x + 70*x^2 + 1274*x^3 + 1333*x^4 - 2109*x^5 - 462*x^6 + 8858*x^7 - 17006*x^8 + 15166*x^9 - 6838*x^10 + 1478*x^11 - 650*x^12 + 760*x^13 - 376*x^14 + 64*x^15)/(1-x)^9. - Vaclav Kotesovec, Mar 25 2010

A173214 Number of ways to place 4 nonattacking amazons (superqueens) on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 2, 112, 1754, 13074, 63400, 234014, 712248, 1882132, 4457246, 9679760, 19584514, 37367934, 67849336, 118085614, 198107620, 321870956, 508359070, 782972820, 1179105738, 1740089734, 2521359260, 3593085246, 5043058972

Offset: 1

Vaclav Kotesovec, Feb 12 2010

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Wikipedia, Amazon (chess)

Cf. A172200, A172201, A061994.

CoefficientList[Series[2 x^4 (28 x^17 - 18 x^16 - 162 x^15 - 139 x^14 + 261 x^13 + 1268 x^12 + 2387 x^11 + 1220 x^10 - 5937 x^9 - 18637 x^8 - 30086 x^7 - 31557 x^6 - 23251 x^5 - 11716 x^4 - 3859 x^3 - 708 x^2 - 53 x - 1) / ((x + 1)^4 (x - 1)^9 (x^2 + x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, May 30 2013 *)

a(n) = n^8/24-5n^7/6+47n^6/9+43n^5/10-5053n^4/24+112585n^3/108-15433n^2/8+55669n/270+119917/54 + (n^3/4-21n^2/8+7n-3/2)*(-1)^n + 32/27*(n-1)*cos(2*Pi*n/3) + 40*sqrt(3)*sin(2*Pi*n/3)/81, n>=6.
Recurrence: a(n) = 3a(n-1)+a(n-2)-9a(n-3)+12a(n-5)+7a(n-6)-15a(n-7)-16a(n-8)+16a(n-9)+15a(n-10)-7a(n-11)-12a(n-12)+9a(n-14)-a(n-15)-3a(n-16)+a(n-17), n>=23. - Vaclav Kotesovec, Feb 18 2010
G.f.: 2x^5*(28x^17-18x^16-162x^15-139x^14+261x^13+1268x^12+2387x^11+1220x^10-5937x^9-18637x^8-30086x^7-31557x^6-23251x^5-11716x^4-3859x^3-708x^2-53x-1)/((x+1)^4*(x-1)^9*(x^2+x+1)^2). - Vaclav Kotesovec, Mar 24 2010

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).

A178717 Degree of denominator of GF for number of ways to place k nonattacking queens on an n X n board.

Original entry on oeis.org

3, 5, 9, 17, 37, 81, 197, 477, 1197, 3077, 7989, 20649, 53885, 140601, 366917, 959685, 2511477, 6571681, 17202449, 45027677, 117871345, 308581637, 807852685, 2114904397, 5536838045, 14495554593, 37949503089, 99352690141, 260108204933

Offset: 1

Vaclav Kotesovec, Jun 07 2010

nonn

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013

Cf. A036464, A047659, A061994, A108792, A176186, A000010, A000045.

Table[2*k + 1 + Sum[Sum[2*j*EulerPhi[i], {i, Fibonacci[k - j] + 1, Fibonacci[k - j + 1]}], {j, 1, k - 1}], {k, 1, 20}]

Explicit formula (Vaclav Kotesovec, May 31 2010), for k>1 : d(k) = 2*k+1+Sum[Sum[2*j*EulerPhi[i],{i,Fibonacci[k-j]+1,Fibonacci[k-j+1]}],{j,1,k-1}].

Asymptotic formula: d(k) ~ 6/(5*Pi^2)*((1+Sqrt[5])/2)^(2*k+1) or d(k) ~ 3*(1+Sqrt[5])/Pi^2*Fibonacci[k]^2.

A190396 Number of ways to place 4 nonattacking grasshoppers on a chessboard of size n x n.

Original entry on oeis.org

0, 1, 78, 1278, 10002, 50191, 189208, 584958, 1563488, 3737987, 8181786, 16669638, 32003238, 58438623, 102234772, 172344406, 281269668, 446107043, 689807558, 1042679982, 1544166426, 2244921423, 3209227248, 4517779918

Offset: 1

Vaclav Kotesovec, May 10 2011

The Grasshopper moves on the same lines as a queen, but must jump over a hurdle to land on the square immediately beyond.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights (in English and Czech)

Cf.: A190395, A061994.

CoefficientList[Series[- x (2 x^9 - 22 x^8 + 50 x^7 + 78 x^6 - 89 x^5 - 245 x^4 + 1224 x^3 + 612 x^2 + 69 x + 1) / (x - 1)^9, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 02 2013 *)

a(n) = 1/24*(n^8 -6*n^6 -80*n^5 +431*n^4 -552*n^3 -666*n^2 +2168*n -1392), n>2.

G.f.: -x^2*(2*x^9 -22*x^8 +50*x^7 +78*x^6 -89*x^5 -245*x^4 +1224*x^3 +612*x^2 +69*x +1)/(x-1)^9.

A252593 Number of ways to place 8 nonattacking queens on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 92, 13848, 636524, 14803480, 207667564, 2008758532, 14752426528, 87154016752, 432539436508, 1858901487620

Offset: 1

Antal Pinter, Dec 18 2014

Conjectured recurrence order is 477 (see "Non-attacking chess pieces", p. 19). - Vaclav Kotesovec, Dec 19 2014

S. Chaiken, C. R. H. Hanusa and T. Zaslavsky, A q-Queens Problem, I. General theory, arXiv:1303.1879 [math.CO], 2013-2014.
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013
Antal Pinter, Combinatorics, software for enumerating positions of non-attacking chess pieces.
I. Rivin, I. Vardi and P. Zimmermann, The n-queens problem, Amer. Math. Monthly, 101 (1994), 629-639.

Cf. A000170, A036464, A047659, A061994, A108792, A176186, A178721.

Column k=8 of A348129.

a(n) = n^16/40320 - n^15/432 + 221*n^14/2160 + O(n^13). - Vaclav Kotesovec, Dec 19 2014

a(16) from Vaclav Kotesovec, Dec 19 2014

a(17) from Vaclav Kotesovec, Dec 20 2014

A269133 Number of ways to place m nonattacking queens on an m X n board, 1 <= m <= n (triangular array).

Original entry on oeis.org

1, 2, 0, 3, 2, 0, 4, 6, 4, 2, 5, 12, 14, 12, 10, 6, 20, 36, 46, 40, 4, 7, 30, 76, 140, 164, 94, 40, 8, 42, 140, 344, 568, 550, 312, 92, 9, 56, 234, 732, 1614, 2292, 2038, 1066, 352, 10, 72, 364, 1400, 3916, 7552, 9632, 7828, 4040, 724, 11, 90, 536, 2468, 8492, 21362, 37248, 44148, 34774, 15116, 2680, 12, 110, 756, 4080, 16852, 52856, 120104, 195270, 222720, 160964, 68264, 14200

Offset: 1

Marko Riedel, Feb 19 2016

			The triangular array begins:
   n\m  1   2   3    4     5     6      7      8      9     10    11    12
   1    1
   2    2   0
   3    3   2   0
   4    4   6   4    2
   5    5  12  14   12    10
   6    6  20  36   46    40     4
   7    7  30  76  140   164    94     40
   8    8  42 140  344   568   550    312     92
   9    9  56 234  732  1614  2292   2038   1066    352
  10   10  72 364 1400  3916  7552   9632   7828   4040    724
  11   11  90 536 2468  8492 21362  37248  44148  34774  15116  2680
  12   12 110 756 4080 16852 52856 120104 195270 222720 160964 68264 14200
...

Math StackExchange, State space for eight queen problem
Marko Riedel, Perl program to compute triangular array of nonattacking queens configurations

Cf. A000170 (m=n), A002562, A065256, A348129.
Cf. A000027 (m=1), A002378 (m=2), A061989 (m=3), A061990 (m=4), A061991 (m=5), A061992 (m=6), A061993 (m=7), A172449 (m=8).
Cf. A036464 (2Q), A047659 (3Q), A061994 (4Q), A108792 (5Q), A176186 (6Q).
Cf. A006717, A051906, A319284 (backtrack trees).

{A269133(m, n, B=[], t=if(#B, setminus(n, Set(concat(B+t=[-#B..-1], B-t))), n=[1..n]))= if(#B < m-1, vecsum([A269133(m, setminus(n, [t]), concat(B,t)) | t<-t]), #t)} \\ M. F. Hasler, Jan 11 2022

cp's OEIS Frontend

A172139 Number of ways to place 4 nonattacking zebras on an n X n board.

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A173214 Number of ways to place 4 nonattacking amazons (superqueens) on an n X n board.

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

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A178717 Degree of denominator of GF for number of ways to place k nonattacking queens on an n X n board.

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A190396 Number of ways to place 4 nonattacking grasshoppers on a chessboard of size n x n.

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A252593 Number of ways to place 8 nonattacking queens on an n X n board.

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A269133 Number of ways to place m nonattacking queens on an m X n board, 1 <= m <= n (triangular array).

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