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A176186 Number of ways to place 6 nonattacking queens on an n X n board. 12
4, 832, 22708, 312956, 2716096, 17117832, 84871680, 349093856, 1239869972, 3905117168, 11139611892, 29224290600, 71402912960, 164029487484, 357164398040, 741835920276, 1477798367368, 2836053660668, 5263672510684, 9478352925488, 16606678238496, 28378012168908 (list; graph; refs; listen; history; text; internal format)
OFFSET
6,1
LINKS
S. Chaiken, C. R. H. Hanusa and T. Zaslavsky, A q-queens problem I. General theory, preprint, August 7, 2014. - From N. J. A. Sloane, Feb 16 2013
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853 [math.CO], 2016-2020.
The formula below was obtained during the programming competition on my blog (in Russian). [From Artem M. Karavaev, May 10 2010]
FORMULA
Contribution from Artem M. Karavaev, May 10 2010: (Start)
a(n)=1/720*n^12-5/72*n^11+77/48*n^10-73339/3240*n^9
+312607/1440*n^8-66917/45*n^7+2226017/300*n^6
-9149222687/340200*n^5+102550276811/1488375*n^4
-2786721974671/23814000*n^3+453909010753/3969000*n^2
-2166093922711/47628000*n+1/432*(72*n^7-2916*n^6
+50580*n^5-485172*n^4+2750088*n^3-8977638*n^2+14596860*n
-7467833)*floor(1/2*n)+2/243*(216*n^5-6480*n^4+81308*n^3
-525474*n^2+1650126*n-1364199)*floor(1/3*n)
+1/243*(216*n^5-6750*n^4+88940*n^3-620184*n^2+2250534*n-3138465)*floor(1/3*n+1/3)
+1/4*(116*n^3-2133*n^2+12902*n-13728)*floor(1/4*n)
+1/4*(100*n^3-1984*n^2+13696*n-26993)*floor(1/4*n+1/4)
-1/4*(16*n^3-149*n^2-794*n+13265)*floor(1/4*n+1/2)
+8/125*(100*n^3-1778*n^2+16360*n-19739)*floor(1/5*n)
+4/125*(150*n^3-2842*n^2+27521*n-65479)*floor(1/5*n+1/5)
+4/125*(100*n^3-1978*n^2+19118*n-54269)*floor(1/5*n+2/5)
+4/125*(50*n^3-1014*n^2+10311*n-30319)*floor(1/5*n+3/5)
+2/27*(2118*n-4499)*floor(1/6*n)
+1/9*(1934*n-6633)*floor(1/6*n+1/6)
+2/27*(1566*n-10901)*floor(1/6*n+1/3)
-4/9*(92*n+1067)*floor(1/6*n+1/2)
-1/27*(2670*n+1903)*floor(1/6*n+2/3)
+48/49*(95*n-173)*floor(1/7*n)
+4/49*(920*n-3597)*floor(1/7*n+1/7)
+8/49*(335*n-1672)*floor(1/7*n+2/7)
+4/49*(600*n-3409)*floor(1/7*n+3/7)
+8/49*(145*n-926)*floor(1/7*n+4/7)
+4/49*(160*n-989)*floor(1/7*n+5/7)
+2*(2*n-5)*floor(1/8*n)
+2*(2*n-9)*floor(1/8*n+1/8)
+2*(2*n-11)*floor(1/8*n+1/4)
+2*(2*n-11)*floor(1/8*n+3/8)
-12*floor(1/8*n+1/2)
-4*floor(1/8*n+5/8).
gf.:-4*x^6*(1+213*x+53192307*x^6+41638044492*x^11+6730*x^2+1148407*x^4
+109349*x^3+8814849*x^5+264701695*x^7+1124196463*x^8+4178943637*x^9
+13860639977*x^10+114647411058*x^12+65980784446603*x^21+39390545501971*x^20
+22595990341656*x^19+12422932793397*x^18+6526629468148*x^17
+3265443398940*x^16+1549727363371*x^15+694388440836*x^14
+106432971812268*x^22+165658741321711*x^23+249214616002036*x^24
+362914602952313*x^25+2587299110418159*x^33+2935747117591644*x^34
+3249296395578274*x^35+3508939440356435*x^36+3697993190167173*x^37
+3803816547345336*x^38+3819162259356822*x^39+3580483108767024*x^41
+512246494510867*x^26+701616116303232*x^27+933484654834212*x^28
+1207502931973542*x^29+1519776851512400*x^30+1862408041738532*x^31
+2223440299348897*x^32+3045052266009747*x^43+2706059061856895*x^44
+2345222492958126*x^45+1981371771386641*x^46+1631097942913736*x^47
+1307636222431031*x^48+1020253102328739*x^49+774143005387556*x^50
+570765121032393*x^51+408502373125992*x^52+283498964529980*x^53
+190537440843487*x^54+123839006896190*x^55+77709213130439*x^56
+46991003436040*x^57+27324795064304*x^58+15241746945993*x^59
+8132464535507*x^60+4137125169063*x^61+1998986925515*x^62
+913299307705*x^63+392473060699*x^64+157632673869*x^65+58720300345*x^66
+20096128176*x^67+6243623123*x^68+1733735219*x^69+421322394*x^70
+86868206*x^71+14475708*x^72+1769106*x^73+125388*x^74+292139023877*x^13
+3742970026288202*x^40+3342672506335632*x^42)
/(x-1)^13/(x+1)^8/(x^2+x+1)^6/(x^2+1)^4/(x^4+x^3+x^2+x+1)^4
/(x^2-x+1)^2/(x^6+x^5+x^4+x^3+x^2+x+1)^2/(x^4+1)^2.
Recurrence: a(0) = 0, a(1) = 0, a(2) = 0, a(3) = 0, a(4) = 0, a(5) = 0,
a(6) = 4,
a(7) = 832,
a(8) = 22708,
a(9) = 312956,
a(10) = 2716096,
a(11) = 17117832,
a(12) = 84871680,
a(13) = 349093856,
a(14) = 1239869972,
a(15) = 3905117168,
a(16) = 11139611892,
a(17) = 29224290600,
a(18) = 71402912960,
a(19) = 164029487484,
a(20) = 357164398040,
a(21) = 741835920276,
a(22) = 1477798367368,
a(23) = 2836053660668,
a(24) = 5263672510684,
a(25) = 9478352925488,
a(26) = 16606678238496,
a(27) = 28378012168908,
a(28) = 47398421913600,
a(29) = 77522788818316,
a(30) = 124365738451680,
a(31) = 195977208395580,
a(32) = 303748457927000,
a(33) = 463582807382736,
a(34) = 697434075907504,
a(35) = 1035256352634420,
a(36) = 1517521355687872,
a(37) = 2198354851112760,
a(38) = 3149525540545556,
a(39) = 4465340754179496,
a(40) = 6268789672000200,
a(41) = 8718985543275112,
a(42) = 12020393279930400,
a(43) = 16433877629761792,
a(44) = 22290259302807700,
a(45) = 30006374870365136,
a(46) = 40104595602917300,
a(47) = 53235736336244300,
a(48) = 70206658745546392,
a(49) = 92012392107526748,
a(50) = 119874540287319196,
a(51) = 155285624835663096,
a(52) = 200061731591400456,
a(53) = 256402868739653996,
a(54) = 326964160566718884,
a(55) = 414936937933564876,
a(56) = 524143826614930088,
a(57) = 659146400314780240,
a(58) = 825370726096096944,
a(59) = 1029248723087783480,
a(60) = 1278382175175730400,
a(61) = 1581726455456457436,
a(62) = 1949802708715765760,
a(63) = 2394934394370749612,
a(64) = 2931519277634600260,
a(65) = 3576331321283597364,
a(66) = 4348866396076652064,
a(67) = 5271724407012487004,
a(68) = 6371045245979662116,
a(69) = 7676988784016499040,
a(70) = 9224280528369282176,
a(71) = 11052810248193489712,
a(72) = 13208310203912865056,
a(73) = 15743096674420482872,
a(74) = 18716907492532573788,
a(75) = 22197814778572880876,
a(76) = 26263252803050925204,
a(77) = 31001134787399966700,
a(78) = 36511107035656245460,
a(79) = 42905907681877306448,
a(80) = 50312888556807094572,
a(n) = a(n-81)+5*a(n-80)+13*a(n-79)+21*a(n-78)+19*a(n-77)
-5*a(n-76)-57*a(n-75)-127*a(n-74)-184*a(n-73)-180*a(n-72)
-70*a(n-71)+162*a(n-70)+476*a(n-69)+768*a(n-68)
+889*a(n-67)+695*a(n-66)+114*a(n-65)-794*a(n-64)
-1806*a(n-63)-2570*a(n-62)-2701*a(n-61)-1929*a(n-60)
-234*a(n-59)+2072*a(n-58)+4374*a(n-57)+5898*a(n-56)
+5950*a(n-55)+4180*a(n-54)+771*a(n-53)-3521*a(n-52)
-7530*a(n-51)-9994*a(n-50)-9959*a(n-49)-7119*a(n-48)
-1994*a(n-47)+4156*a(n-46)+9657*a(n-45)+12909*a(n-44)
+12881*a(n-43)+9447*a(n-42)+3464*a(n-41)-3464*a(n-40)
-9447*a(n-39)-12881*a(n-38)-12909*a(n-37)-9657*a(n-36)
-4156*a(n-35)+1994*a(n-34)+7119*a(n-33)+9959*a(n-32)
+9994*a(n-31)+7530*a(n-30)+3521*a(n-29)-771*a(n-28)
-4180*a(n-27)-5950*a(n-26)-5898*a(n-25)-4374*a(n-24)
-2072*a(n-23)+234*a(n-22)+1929*a(n-21)+2701*a(n-20)
+2570*a(n-19)+1806*a(n-18)+794*a(n-17)-114*a(n-16)
-695*a(n-15)-889*a(n-14)-768*a(n-13)-476*a(n-12)
-162*a(n-11)+70*a(n-10)+180*a(n-9)+184*a(n-8)
+127*a(n-7)+57*a(n-6)+5*a(n-5)-19*a(n-4)-21*a(n-3)
-13*a(n-2)-5*a(n-1), n>80.
(End)
MATHEMATICA
(* Alternative formula by Vaclav Kotesovec *)
q6nn = 1/720*n^12-5/72*n^11+77/48*n^10-73339/3240*n^9+312727/1440*n^8-268283/180*n^7+26932229/3600*n^6-18719233399/680400*n^5+577434913967/7938000*n^4-117739826734/893025*n^3+112056711821/756000*n^2-393833819123/4762800*n+749037381271/68040000\
+(1/24*n^7-27/16*n^6+1405/48*n^5-13477/48*n^4+114587/72*n^3-1496273/288*n^2+1216405/144*n-7467833/1728)*(-1)^n\
+(25*n^3/4-124*n^2+856*n-26993/16)*Sqrt[2]*Cos[Pi*n/2+Pi/4]\
+(33*n^3/4-1141*n^2/8+3027*n/4-463/16)*Sqrt[2]*Sin[Pi*n/2+Pi/4]\
+2/243*((383792n-552926)*Cos[Pi*n/3]\
+(493229-366386*n)*Cos[Pi*n/3+Pi/3]\
+(531641-363074*n)*Sin[Pi*n/3+Pi/6])\
+2*(366386*n-493229)*Sqrt[3]*Cos[2*Pi*n/3+Pi/6]/729\
-(16*n^5/81-500*n^4/81+177880*n^3/2187-413456*n^2/729+767584*n/729-1105852/729)*Sqrt[3]*Sin[2*Pi*n/3]\
+(32*n^5/81-320*n^4/27+325232*n^3/2187-77848*n^2/81+18116*n/9-832474/729)*Sqrt[3]*Sin[2*Pi*n/3+Pi/3]\
+(2*n-11)/2*(Sqrt[2]+1)*Cos[Pi*n/4+Pi/4]\
+(Sqrt[4+2*Sqrt[2]]/4)*((2*n+1)*Sin[Pi*n/4+Pi/8]+(2*n-7)*Sin[Pi*n/4+3*Pi/8])\
+(2*n-11)/2*(Sqrt[2]-1)*Sin[3*Pi*n/4+Pi/4]\
+(Sqrt[4-2*Sqrt[2]]/4)*((7-2*n)*Sin[3*Pi*n/4+Pi/8]+(2*n+1)*Sin[3*Pi*n/4+3*Pi/8])\
+((4*Sqrt[50+10*Sqrt[5]])/3125)*((150*n^3-2842*n^2+27521*n-65479)*Cos[2*Pi*n/5+Pi/10]\
-(100*n^3-1978*n^2+19118*n-54269)*Sin[2*Pi*n/5]\
+2*(100*n^3-1778*n^2+16360*n-19739)*Sin[2*Pi*n/5+Pi/5]\
-(50*n^3-1014*n^2+10311*n-30319)*Sin[2*Pi*n/5+2*Pi/5])\
+((4*Sqrt[50-10*Sqrt[5]])/3125)*((50*n^3-1014*n^2+10311*n-30319)*Cos[4*Pi*n/5+3*Pi/10]\
+(100*n^3-1978*n^2+19118*n-54269)*Sin[4*Pi*n/5]\
-(150*n^3-2842*n^2+27521*n-65479)*Sin[4*Pi*n/5+Pi/5]\
+2*(100*n^3-1778*n^2+16360*n-19739)*Sin[4*Pi*n/5+2*Pi/5])\
+4/(343*Sin[Pi/7])*((989-160*n)*Cos[2*Pi*n/7+Pi/14]\
+2*(335*n-1672)*Cos[2*Pi*n/7+3*Pi/14]\
+(3409-600*n)*Sin[2*Pi*n/7]\
+12*(95*n-173)*Sin[2*Pi*n/7+Pi/7]\
+2*(926-145*n)*Sin[2*Pi*n/7+2*Pi/7]\
+(920*n-3597)*Sin[2*Pi*n/7+3*Pi/7])\
+4/(343*Cos[3*Pi/14])*(2*(145*n-926)*Cos[4*Pi*n/7+Pi/14]\
+(920*n-3597)*Cos[4*Pi*n/7+5*Pi/14]\
+(600*n-3409)*Sin[4*Pi*n/7]\
+(989-160*n)*Sin[4*Pi*n/7+Pi/7]\
+12*(95*n-173)*Sin[4*Pi*n/7+2*Pi/7]\
+2*(1672-335*n)*Sin[4*Pi*n/7+3*Pi/7])\
+4/(343*Cos[Pi/14])*((160*n-989)*Cos[6*Pi*n/7+3*Pi/14]\
+2*(926-145*n)*Cos[6*Pi*n/7+5*Pi/14]\
+(3409-600*n)*Sin[6*Pi*n/7]\
+(3597-920*n)*Sin[6*Pi*n/7+2*Pi/7]\
+12*(95*n-173)*Sin[6*Pi*n/7+3*Pi/7]\
+2*(335*n-1672)*Sin[6*Pi*n/7+Pi/7])
CROSSREFS
Cf. A108792.
Column k=6 of A348129.
Sequence in context: A172893 A072725 A349067 * A332184 A221232 A272167
KEYWORD
nonn,nice,easy
AUTHOR
Artem M. Karavaev, Apr 11 2010
STATUS
approved

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Last modified February 25 16:57 EST 2024. Contains 370332 sequences. (Running on oeis4.)