A061992 -id:A061992 - cp's OEIS frontend

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

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.

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
```
# See the link section.
```

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

A172205 Number of ways to place 6 nonattacking kings on a 6 X n board.

Original entry on oeis.org

0, 0, 16, 408, 8544, 62266, 291908, 1021254, 2916232, 7179314, 15790572, 31795390, 59638832, 105546666, 177953044, 287974838, 449932632, 681918370, 1006409660, 1450930734, 2048760064, 2839684634, 3870800868, 5197362214, 6883673384

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. A172158, A061992, A172202, A172203, A172204.

CoefficientList[Series[-2 x^2 (475 x^8 - 1015 x^7 + 4398 x^6 + 194 x^5 + 10875 x^4 + 5233 x^3 + 3012 x^2 + 148 x + 8) / (x - 1)^7, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)

a(n) = 2*(162n^6-3240n^5+29160n^4-151830n^3+483798n^2-895085n+749335)/5, n>=5.

G.f.: -2*x^3*(475*x^8-1015*x^7+4398*x^6+194*x^5+10875*x^4+5233*x^3+3012*x^2 +148*x+8)/(x-1)^7. - Vaclav Kotesovec, Mar 24 2010

A172215 Number of ways to place 6 nonattacking knights on a 6 X n board.

Original entry on oeis.org

1, 58, 729, 8830, 60285, 257318, 858262, 2404448, 5879329, 12927182, 26115008, 49238436, 87675623, 148787822, 242366502, 381127124, 581249573, 862965246, 1251190796, 1776208532, 2474393475, 3388987070, 4570917554, 6079666980

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 (7, -21, 35, -35, 21, -7, 1).

Cf. A061992, A172212, A172213, A172214.

CoefficientList[Series[-(104 x^15 - 116 x^14 - 1328 x^13 + 3992 x^12 + 806 x^11 - 16380 x^10 + 27343 x^9 - 4845 x^8 - 15537 x^7 + 38275 x^6 - 2753 x^5 + 11789 x^4 + 4910 x^3 + 344 x^2 + 51 x + 1) / (x - 1)^7, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,58,729,8830,60285,257318,858262,2404448,5879329,12927182,26115008,49238436,87675623,148787822,242366502,381127124},30] (* Harvey P. Dale, Dec 31 2022 *)

a(n) = (648n^6-11340n^5+103770n^4-606645n^3+2328317n^2-5466660n+6051720)/10, n>=10.
G.f.: -x * (104*x^15 -116*x^14 -1328*x^13 +3992*x^12 +806*x^11 -16380*x^10 +27343*x^9 -4845*x^8 -15537*x^7 +38275*x^6 -2753*x^5 +11789*x^4 +4910*x^3 +344*x^2 +51*x +1) / (x-1)^7. - Vaclav Kotesovec, Mar 25 2010
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). - Wesley Ivan Hurt, Apr 16 2023

A172232 Number of ways to place 6 nonattacking wazirs on a 6 X n board.

Original entry on oeis.org

0, 2, 504, 10010, 78052, 368868, 1280832, 3612344, 8774380, 19049692, 37898664, 70311824, 123209012, 205885204, 330502992, 512631720, 771833276, 1132294540, 1623506488, 2280989952, 3147068036, 4271685188, 5713272928, 7539662232, 9829042572, 12670967612

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. A172229, A172230, A172231, A061992.

CoefficientList[Series[- 2 x (3 x^9 - 5 x^8 + 100 x^7 + 354 x^6 + 2548 x^5 + 7572 x^4 + 9248 x^3 + 3262 x^2 + 245 x + 1) / (x - 1)^7, {x, 0, 50}], x] (* Vincenzo Librandi, May 28 2013 *)

a(n) = 2*(486*n^6 -5670*n^5 +30240*n^4 -95230*n^3 +187899*n^2 -220775*n +120540) / 15, n>=5.

G.f.: -2*x^2 * (3*x^9 -5*x^8 +100*x^7 +354*x^6 +2548*x^5 +7572*x^4 +9248*x^3 +3262*x^2 +245*x +1) / (x-1)^7. - Vaclav Kotesovec, Mar 25 2010

A172211 Number of ways to place 6 nonattacking bishops on a 6 X n board.

Original entry on oeis.org

1, 16, 313, 2320, 12160, 53744, 209428, 683524, 1905625, 4664384, 10297579, 20907590, 39664250, 71114916, 121559433, 199459466, 315906248, 485124352, 725031335, 1057839684, 1510706686, 2116429956, 2914190277, 3950340692

Offset: 1

Vaclav Kotesovec, Jan 29 2010

Vaclav Kotesovec and Vincenzo Librandi, Table of n, a(n) for n = 1..1000 (first 32 terms from V. Kotesovec)
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes

Cf. A061992, A172207, A172208, A172210.

CoefficientList[Series[-(2 x^30 - 6 x^29 + 14 x^28 - 26 x^27 + 44 x^26 - 220 x^25 + 596 x^24 - 1060 x^23 + 1654 x^22 - 2266 x^21 + 5622 x^20 - 13570 x^19 + 19848 x^18 - 22392 x^17 + 24048 x^16 - 30525 x^15 + 57673 x^14 - 80154 x^13 + 61962 x^12 - 30874 x^11 + 25832 x^10 - 9360 x^9 + 16960 x^8 - 4710 x^7 + 18006 x^6 + 6928 x^5 + 1968 x^4 + 430 x^3 + 222 x^2 + 9 x + 1) / (x - 1)^7, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)

a(n) = (648n^6-17820n^5+240930n^4-2011545n^3+10806047n^2-35094560n+53430940)/10, n>=25.
For any fixed value of k > 1, a(n) = 1/k!*(kn)^k - (2k-1)/2/(k-2)!*(kn)^(k-1) + ...
G.f.: -x*(2*x^30-6*x^29+14*x^28-26*x^27+44*x^26-220*x^25+596*x^24-1060*x^23+1654*x^22
-2266*x^21+5622*x^20-13570*x^19+19848*x^18-22392*x^17+24048*x^16-30525*x^15+57673*x^14
-80154*x^13+61962*x^12-30874*x^11+25832*x^10-9360*x^9+16960*x^8-4710*x^7+18006*x^6+6928*x^5
+1968*x^4+430*x^3+222*x^2+9*x+1)/(x-1)^7. - Vaclav Kotesovec, Mar 25 2010

A172224 Number of ways to place 6 nonattacking zebras on a 6 X n board.

Original entry on oeis.org

1, 924, 8989, 37270, 145233, 525796, 1605490, 4136952, 9435413, 19632414, 37957424, 69050898, 119351315, 197524064, 314935542, 486171662, 729604121, 1068003424, 1529198580, 2146783422, 2960869583, 4018886128, 5376425842

Offset: 1

Vaclav Kotesovec, Jan 29 2010

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

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, Zebra Graph
Wikipedia, Zebra (chess)

Cf. A061992, A172221, A172222, A172223.

CoefficientList[Series[-(32 x^20 - 48 x^19 - 84 x^18 - 1004 x^17 + 3350 x^16 - 802 x^15 + 3364 x^14 - 32132 x^13 + 42540 x^12 + 3538 x^11 + 10674 x^10 - 126767 x^9 + 151663 x^8 - 20769 x^7 - 34421 x^6 + 9539 x^5 + 40807 x^4 - 6284 x^3 + 2542 x^2 + 917 x + 1) / (x - 1)^7, {x, 0, 50}], x] (* Vincenzo Librandi, May 28 2013 *)

a(n) = (1944n^6-27540n^5+227070n^4-1222555n^3+4366071n^2-9580580n+9925860)/30, n>=15.
For any fixed value of k > 1, a(n) = 1/k!*(kn)^k - (k-1)(9k-20)/2/k!*(kn)^(k-1) + ...
G.f.: -x * (32*x^20 -48*x^19 -84*x^18 -1004*x^17 +3350*x^16 -802*x^15 +3364*x^14 -32132*x^13 +42540*x^12 +3538*x^11 +10674*x^10 -126767*x^9 +151663*x^8 -20769*x^7 -34421*x^6 +9539*x^5 +40807*x^4 -6284*x^3 +2542*x^2 +917*x +1) / (x-1)^7. - Vaclav Kotesovec, Mar 25 2010

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

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A061993 Number of ways to place 7 nonattacking queens on a 7 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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A172449 Number of ways to place 8 nonattacking queens on an 8 X n board.

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A172205 Number of ways to place 6 nonattacking kings on a 6 X n board.

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A172215 Number of ways to place 6 nonattacking knights on a 6 X n board.

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A172232 Number of ways to place 6 nonattacking wazirs on a 6 X n board.

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A172211 Number of ways to place 6 nonattacking bishops on a 6 X n board.

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A172224 Number of ways to place 6 nonattacking zebras on a 6 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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