A172202 -id:A172202 - cp's OEIS frontend

A172212 Number of ways to place 3 nonattacking knights on a 3 X n board.

1, 12, 36, 100, 233, 456, 796, 1280, 1935, 2788, 3866, 5196, 6805, 8720, 10968, 13576, 16571, 19980, 23830, 28148, 32961, 38296, 44180, 50640, 57703, 65396, 73746, 82780, 92525, 103008, 114256, 126296, 139155, 152860, 167438, 182916, 199321

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. A172134, A061989, A047659, A172202.

CoefficientList[Series[(6 x^6 - 8 x^5 + 2 x^4 + 24 x^3 - 6 x^2 + 8 x + 1) / (x - 1)^4, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)

a(n) = (9n^3 - 45n^2 + 122n - 144)/2, n>=4.

G.f.: x*(6*x^6-8*x^5+2*x^4+24*x^3-6*x^2+8*x+1)/(x-1)^4. - Vaclav Kotesovec, Mar 25 2010

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

A172204 Number of ways to place 5 nonattacking kings on a 5 X n board.

Original entry on oeis.org

0, 0, 15, 194, 1974, 9856, 34475, 95466, 224589, 468854, 893646, 1585850, 2656976, 4246284, 6523909, 9693986, 13997775, 19716786, 27175904, 36746514, 48849626, 63959000, 82604271, 105374074, 132919169

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. A061998, A061991, A172202, A172203.

CoefficientList[Series[x^2 * (259 * x^6 - 204 * x^5 + 1294 * x^4 + 622 * x^3 + 1035 * x^2 + 104 * x + 15)/(x - 1)^6, {x, 0, 40}], x] (* Vincenzo Librandi, May 27 2013 *)

a(n) = (625n^5-9750n^4+66415n^3-247626n^2+504664n-446544)/24, n>=4.

G.f.: x^3*(259*x^6-204*x^5+1294*x^4+622*x^3+1035*x^2+104*x+15)/(x-1)^6. - Vaclav Kotesovec, Mar 24 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

A172206 Number of ways to place 7 nonattacking kings on a 7 X n board.

Original entry on oeis.org

0, 0, 24, 926, 37282, 394202, 2484382, 10999618, 38168864, 110899878, 281638602, 643766432, 1352358921, 2651129458, 4906381466, 8648792662, 14623854922, 23851793294, 37697787702, 57953320884, 86929476107, 127563008202, 183536011462, 259410007946, 360775279732

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. A061993, A172202, A172203, A172204, A172205.

CoefficientList[Series[x^2 (3387 x^10 - 13990 x^9 + 57102 x^8 - 55038 x^7 + 217860 x^6 + 137902 x^5 + 324486 x^4 + 120530 x^3 + 30546 x^2 + 734 x + 24) / (x - 1)^8, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)

a(n) = (117649n^7 -2873997n^6 +32197753n^5 -215350695n^4 +932130286n^3 -2618213868n^2 +4424623272n -3468569760)/720, n>=6. For any fixed value of k > 1, a(n) = 1/k!*(kn)^k - 3(k-1)(3k-2)/2/k!*(kn)^(k-1) + ... .

G.f.: x^3*(3387*x^10 -13990*x^9 +57102*x^8 -55038*x^7 +217860*x^6 +137902*x^5 +324486*x^4 +120530*x^3 +30546*x^2 +734*x +24)/(x-1)^8. - Vaclav Kotesovec, Mar 24 2010

A172261 Number of ways to place 8 nonattacking kings on an 8 X n board.

Original entry on oeis.org

0, 0, 25, 1847, 162531, 2501726, 21243084, 119138166, 502726650, 1724809105, 5059647669, 13132889249, 30905051345, 67124176002, 136380034610, 261909043488, 479315827404, 841394145399, 1424246670499, 2334919892115

Offset: 1

Vaclav Kotesovec, Jan 30 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. A172202, A172203, A172204, A172205, A172206.

CoefficientList[Series[- x^2 (11814 x^12 - 80082 x^11 + 366204 x^10 - 759794 x^9 + 1916625 x^8 - 283007 x^7 + 5337480 x^6 + 4589514 x^5 + 4426668 x^4 + 1103339 x^3 + 146808 x^2 + 1622 x + 25) / (x - 1)^9, {x, 0, 50}], x] (* Vincenzo Librandi, May 29 2013 *)

a(n) = (1048576n^8 -30277632n^7 +406210560n^6 -3319585920n^5 +18136811049n^4 -68048382318n^3 +171628664735n^2 -266425935930n +194935658400)/2520, n>=7.

G.f.: -x^3*(11814*x^12 -80082*x^11 +366204*x^10 -759794*x^9 +1916625*x^8 -283007*x^7 +5337480*x^6 +4589514*x^5 +4426668*x^4 +1103339*x^3 +146808*x^2 +1622*x +25)/(x-1)^9. [Vaclav Kotesovec, Mar 24 2010]

cp's OEIS Frontend

A172212 Number of ways to place 3 nonattacking knights on a 3 X n board.

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

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A172204 Number of ways to place 5 nonattacking kings on a 5 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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A172206 Number of ways to place 7 nonattacking kings on a 7 X n board.

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A172261 Number of ways to place 8 nonattacking kings on an 8 X n board.

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