A061990 -id:A061990 - cp's OEIS frontend

A061989 Number of ways to place 3 nonattacking queens on a 3 X n board.

0, 0, 0, 0, 4, 14, 36, 76, 140, 234, 364, 536, 756, 1030, 1364, 1764, 2236, 2786, 3420, 4144, 4964, 5886, 6916, 8060, 9324, 10714, 12236, 13896, 15700, 17654, 19764, 22036, 24476, 27090, 29884, 32864, 36036, 39406, 42980, 46764, 50764

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 29 2001

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Ways of placing non-attacking queens and kings..., part of "Between chessboard and computer", 1996, pp. 204 - 206.
E. Lucas, Recreations mathematiques I, Albert Blanchard, Paris, 1992, p. 231.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A061990, A117937, A229183.

Essentially the same as A079908.

[0,0,0] cat [(n-3)*(n^2-6*n+12): n in [3..50]]; // G. C. Greubel, Apr 29 2022

A061989 := proc(n)
    if n >= 3 then
        (n-3)*(n^2-6*n+12) ;
    else
        0;
    end if;
end proc:
seq(A061989(n),n=0..30) ; # R. J. Mathar, Aug 16 2019

CoefficientList[Series[2*x^4*(2-x+2*x^2)/(1-x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, May 02 2013 *)

[0,0,0]+[(n-3)*((n-3)^2 +3) for n in (3..50)] # G. C. Greubel, Apr 29 2022

G.f.: 2*x^4*(2-x+2*x^2)/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 7.
Explicit formula (H. Tarry, 1890): a(n) = (n-3)*(n^2-6*n+12), n >= 3.
(4, 14, 36, ...) is the binomial transform of row 4 of A117937: (4, 10, 12, 6). - Gary W. Adamson, Apr 09 2006
a(n) = 2*A229183(n-3). - R. J. Mathar, Aug 16 2019
E.g.f.: 36 + 14*x + 2*x^2 + (-36 + 22*x - 6*x^2 + x^3)*exp(x). - G. C. Greubel, Apr 29 2022

A061991 Number of ways to place 5 nonattacking queens on a 5 X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 10, 40, 164, 568, 1614, 3916, 8492, 16852, 31100, 54068, 89428, 141812, 216932, 321700, 464348, 654548, 903532, 1224212, 1631300, 2141428, 2773268, 3547652, 4487692, 5618900, 6969308, 8569588, 10453172, 12656372, 15218500, 18181988, 21592508

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 31 2001

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
V. Kotesovec, Ways of placing non-attacking queens and kings..., part of "Between chessboard and computer", 1996, pp. 204 - 206.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A061989, A061990.

CoefficientList[Series[2 x^5 (4 x^11 -11 x^10 + 16 x^9 + 7 x^8 - 32 x^7 + 38 x^6 + 6 x^5 + 8 x^4 - 8 x^3 + 37 x^2 - 10 x + 5) / (x-1)^6, {x, 0, 30}], x] (* Vincenzo Librandi, May 12 2013 *)

G.f.: 2*x^5*(4*x^11 - 11*x^10 + 16*x^9 + 7*x^8 - 32*x^7 + 38*x^6 + 6*x^5 + 8*x^4 - 8*x^3 + 37*x^2 - 10*x + 5)/(x - 1)^6.
Recurrence: a(n) = 6*a(n - 1) - 15*a(n - 2) + 20*a(n - 3) - 15*a(n - 4) + 6*a(n - 5) - a(n - 6), n >= 17.
Explicit formula (V. Kotesovec, 1992): a(n) = n^5 - 30*n^4 + 407*n^3 - 3098*n^2 + 13104*n - 24332, n >= 11.

A061992 Number of ways to place 6 nonattacking queens on a 6 X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 94, 550, 2292, 7552, 21362, 52856, 117694, 241484, 463038, 838816, 1448002, 2398292, 3832374, 5935120, 8941514, 13145292, 18908302, 26670584, 36961170, 50409604, 67758182, 89874912, 117767194, 152596220

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 31 2001

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes, part of V. Kotesovec, Between chessboard and computer, 1996, pp. 204 - 206.

Cf. A061989, A061990, A061991.

CoefficientList[Series[-2 x^6 (4 x^17 -12 x^16 + 12 x^15 + 10 x^14 - 10 x^13 + 40 x^12 - 278 x^11 + 677 x^10 - 582 x^9 - 62 x^8 + 654 x^7 - 501 x^6 + 293 x^5 - 46 x^4 + 138 x^3 - 12 x^2 + 33 x + 2) / (x-1)^7, {x, 0, 40}], x] (* Vincenzo Librandi, May 12 2013 *)

G.f.: - 2*x^6*(4*x^17 - 12*x^16 + 12*x^15 + 10*x^14 - 10*x^13 + 40*x^12 - 278*x^11 + 677*x^10 - 582*x^9 - 62*x^8 + 654*x^7 - 501*x^6 + 293*x^5 - 46*x^4 + 138*x^3 - 12*x^2 + 33*x + 2)/(x - 1)^7.
Recurrence: a(n) = 7*a(n - 1) - 21*a(n - 2) + 35*a(n - 3) - 35*a(n - 4) + 21*a(n - 5) - 7*a(n - 6) + a(n - 7), n >= 24.
Explicit formula (V.Kotesovec, 1992): a(n) = n^6 - 45*n^5 + 943*n^4 - 11755*n^3 + 91480*n^2 - 418390*n + 870920, n >= 17.

A061993 Number of ways to place 7 nonattacking queens on a 7 X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 40, 312, 2038, 9632, 37248, 120104, 335010, 835056, 1897702, 3998456, 7907094, 14818300, 26512942, 45562852, 75580634, 121520020, 190031678, 289879092, 432420154, 632159540, 907376502, 1280833348

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 10 2001

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes, part of V. Kotesovec, Between chessboard and computer, 1996, pp. 204 - 206.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A061989, A061990, A061991, A061992.

CoefficientList[Series[2*x^7*(20-4*x+331*x^2-88*x^3+1292*x^4-1356*x^5+2019*x^6 +264*x^7-2857*x^8+6472*x^9-7616*x^10+7462*x^11-7831*x^12+8326*x^13-5672*x^14 +1998*x^15-308*x^16-142*x^17+510*x^18-284*x^19-220*x^20+320*x^21-140*x^22 +24*x^23)/(1-x)^8, {x, 0, 40}], x] (* Vincenzo Librandi, May 12 2013 *)

def p(x): return 20-4*x+331*x^2-88*x^3+1292*x^4-1356*x^5+2019*x^6 +264*x^7-2857*x^8+6472*x^9-7616*x^10+7462*x^11-7831*x^12+8326*x^13-5672*x^14 +1998*x^15-308*x^16-142*x^17+510*x^18-284*x^19-220*x^20+320*x^21-140*x^22 +24*x^23
[( 2*x^7*p(x)/(1-x)^8 ).series(x,n+1).list()[n] for n in (0..40)] # G. C. Greubel, Apr 29 2022

Explicit formula (V. Kotesovec, 1992): a(n) = n^7 - 63*n^6 + 1879*n^5 - 34411*n^4 + 417178*n^3 - 3336014*n^2 + 16209916*n - 36693996, n >= 23.
Recurrence: a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8), n >= 31.
G.f.: 2*x^7*(20 - 4*x + 331*x^2 - 88*x^3 + 1292*x^4 - 1356*x^5 + 2019*x^6 + 264*x^7 - 2857*x^8 + 6472*x^9 - 7616*x^10 + 7462*x^11 - 7831*x^12 + 8326*x^13 - 5672*x^14 + 1998*x^15 - 308*x^16 - 142*x^17 + 510*x^18 - 284*x^19 - 220*x^20 + 320*x^21 - 140*x^22 + 24*x^23)/(1 - x)^8.

A172213 Number of ways to place 4 nonattacking knights on a 4 X n board.

Original entry on oeis.org

1, 16, 84, 412, 1416, 3640, 7928, 15384, 27352, 45432, 71480, 107608, 156184, 219832, 301432, 404120, 531288, 686584, 873912, 1097432, 1361560, 1670968, 2030584, 2445592, 2921432

Offset: 1

Vaclav Kotesovec, Jan 29 2010

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
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A172135, A061990, A172212.

CoefficientList[Series[-(16 x^9 - 20 x^8 - 40 x^7 + 172 x^6 - 81 x^5 + 41 x^4 + 142 x^3 + 14 x^2 + 11 x + 1) / (x - 1)^5, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{1,16,84,412,1416,3640,7928,15384,27352,45432},30] (* Harvey P. Dale, Apr 16 2022 *)

Explicit formula (Vaclav Kotesovec, 26.1.2010): a(n) = 8*(4*n^4-36*n^3+170*n^2-450*n+537)/3, n>=6.
G.f.: -x*(16*x^9-20*x^8-40*x^7+172*x^6-81*x^5+41*x^4+142*x^3+14*x^2+11*x+1)/(x-1)^5. - Vaclav Kotesovec, Mar 25 2010
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, Jun 26 2025

A319284 The profiles of the backtrack tree for the n queens problem, triangle read by rows.

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Sep 16 2018

The profile (p_0, p_1, ..., p_n) is the number of nodes at each level of the tree.

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

D. E. Knuth, The Art of Computer Programming, Volume 4, Pre-fascicle 5B, Introduction to Backtracking, 7.2.2. Backtrack programming. 2018.

Peter Luschny, Rows n = 0..19, flattened
Candida Bowtell and Peter Keevash, The n-queens problem, arXiv:2109.08083 [math.CO] 2021.
V. Kotesovec, Ways of placing non-attacking queens and kings..., part of "Between chessboard and computer", 1996, pp. 204 - 206.
Peter Luschny, Julia implementation of the n queens problem with profiles
Michael Simkin, The number of n-queens configurations, arXiv:2107.13460 [math.CO] 2021.
Wikipedia, Backtracking
Wikipedia, Eight queens puzzle

Cf. A000170 (T(n,n)), A319283 (row sums), A319288 (indices of the row maxima).

Cf. A000012 (col. 0), A000027 (col. 1), A002378 (col. 2), A061989 and A079908 (col. 3), A061990 (col. 4), A061991 (col. 5), A061992 (col. 6), A061993 (col. 7), A172449 (col. 8).

Julia
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# See the link section.
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A172203 Number of ways to place 4 nonattacking kings on a 4 X n board.

Original entry on oeis.org

0, 0, 9, 79, 454, 1566, 4103, 9009, 17484, 30984, 51221, 80163, 120034, 173314, 242739, 331301, 442248, 579084, 745569, 945719, 1183806, 1464358, 1792159, 2172249, 2609924

Offset: 1

Vaclav Kotesovec, Jan 29 2010

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

Cf. A061997, A061990, A172202.

CoefficientList[Series[- x^2 (68 x^4 - 4 x^3 + 149 x^2 + 34 x + 9) / (x - 1)^5, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)

a(n) = (64n^4 - 720n^3 + 3347n^2 - 7569n + 6894)/6, n>=3.

G.f.: -x^3*(68*x^4-4*x^3+149*x^2+34*x+9)/(x-1)^5. - Vaclav Kotesovec, Mar 24 2010

A172230 Number of ways to place 4 nonattacking wazirs on a 4 X n board.

Original entry on oeis.org

0, 2, 61, 405, 1502, 4072, 9091, 17791, 31660, 52442, 82137, 123001, 177546, 248540, 339007, 452227, 591736, 761326, 965045, 1207197, 1492342, 1825296, 2211131, 2655175, 3163012, 3740482, 4393681, 5128961, 5952930, 6872452, 7894647

Offset: 1

Vaclav Kotesovec, Jan 29 2010

Wazir is a (fairy chess) leaper [0,1].

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
Eric Weisstein's World of Mathematics, Grid Graph
Wikipedia, Wazir (chess)

Cf. A172227, A172229, A061990.

CoefficientList[Series[- x (4 x^5 + 12 x^4 + 67 x^3 + 120 x^2 + 51 x + 2) / (x - 1)^5, {x, 0, 50}], x] (* Vincenzo Librandi, May 28 2013 *)

a(n) = (64*n^4 - 432*n^3 + 1235*n^2 - 1797*n + 1122)/6, n>=3.

G.f.: -x^2*(4*x^5+12*x^4+67*x^3+120*x^2+51*x+2)/(x-1)^5. - Vaclav Kotesovec, Mar 25 2010

More terms from Vincenzo Librandi, May 28 2013

A172449 Number of ways to place 8 nonattacking queens on an 8 X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 92, 1066, 7828, 44148, 195270, 707698, 2211868, 6120136, 15324708, 35312064, 75937606, 153942964, 296590536, 546621416, 968910732, 1659114170, 2754780934, 4449361442, 7009572728, 10796663102, 16292133888

Offset: 1

Vaclav Kotesovec, Feb 03 2010

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

Cf. A061989, A061990, A061991, A061992, A061993.

CoefficientList[Series[x^7 (-72 x^31 + 360 x^30 - 360 x^29 - 1320 x^28 + 4208 x^27 - 9064 x^26 + 28358 x^25 - 65290 x^24 + 80160 x^23 - 41550 x^22 - 19482 x^21 + 62314 x^20 - 43912 x^19 - 81620 x^18 + 228424 x^17 - 261720 x^16 + 248114 x^15 - 336290 x^14 + 460564 x^13 - 453438 x^12 + 288474 x^11 - 135252 x^10 + 80270 x^9 - 85476 x^8 + 49676 x^7 - 23614 x^6 - 4768 x^5 - 1794 x^4 - 4344 x^3 - 1546 x^2 - 238 x - 92) / (x - 1)^9, {x, 0, 50}], x] (* Vincenzo Librandi, May 29 2013 *)

a(n) = n^8 - 84*n^7 + 3378*n^6 - 85078*n^5 + 1467563*n^4 - 17723656*n^3 + 145910074*n^2 - 745654756*n + 1802501048, for n >= 31. - Vaclav Kotesovec, Feb 03 2010

G.f.: x^8*(-72*x^31 + 360*x^30 - 360*x^29 - 1320*x^28 + 4208*x^27 - 9064*x^26 + 28358*x^25 - 65290*x^24 + 80160*x^23 - 41550*x^22 - 19482*x^21 + 62314*x^20 - 43912*x^19 - 81620*x^18 + 228424*x^17 - 261720*x^16 + 248114*x^15 - 336290*x^14 + 460564*x^13 - 453438*x^12 + 288474*x^11 - 135252*x^10 + 80270*x^9 - 85476*x^8 + 49676*x^7 - 23614*x^6 - 4768*x^5 - 1794*x^4 - 4344*x^3 - 1546*x^2 - 238*x - 92)/(x-1)^9. - Vaclav Kotesovec, Mar 20 2010

A172208 Number of ways to place 4 nonattacking bishops on a 4 X n board.

Original entry on oeis.org

1, 9, 61, 260, 927, 2578, 5965, 12066, 22135, 37678, 60457, 92488, 136043, 193650, 268093, 362412, 479903, 624118, 798865, 1008208, 1256467, 1548218, 1888293, 2281780, 2734023, 3250622, 3837433, 4500568, 5246395, 6081538, 7012877

Offset: 1

Vaclav Kotesovec, Jan 29 2010

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
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A172127, A061990, A172207.

CoefficientList[Series[-1 (2 x^12 - 2 x^11 + 4 x^10 - 24 x^9 + 50 x^8 - 10 x^7 + 41 x^6 - 23 x^5 + 152 * x^4 + 35 x^3 + 26  x^2 + 4 x + 1) / (x - 1)^5, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{1,9,61,260,927,2578,5965,12066,22135,37678,60457,92488,136043},40] (* Harvey P. Dale, Dec 13 2021 *)

a(n) = (32*n^4 -336*n^3 +1702*n^2 -4701*n +5844) / 3, n>=9.

G.f.: -x * (2*x^12 -2*x^11 +4*x^10 -24*x^9 +50*x^8 -10*x^7 +41*x^6 -23*x^5 +152*x^4 +35*x^3 +26*x^2 +4*x+1) / (x-1)^5. - Vaclav Kotesovec, Mar 25 2010

cp's OEIS Frontend

A061989 Number of ways to place 3 nonattacking queens on a 3 X n board.

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A061991 Number of ways to place 5 nonattacking queens on a 5 X n board.

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A061992 Number of ways to place 6 nonattacking queens on a 6 X n board.

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A061993 Number of ways to place 7 nonattacking queens on a 7 X n board.

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A172213 Number of ways to place 4 nonattacking knights on a 4 X n board.

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A319284 The profiles of the backtrack tree for the n queens problem, triangle read by rows.

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A172203 Number of ways to place 4 nonattacking kings on a 4 X n board.

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A172230 Number of ways to place 4 nonattacking wazirs on a 4 X n board.

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