A108792 Number of ways to place 5 nonattacking queens on an n X n board.

10, 248, 4618, 46736, 310496, 1535440, 6110256, 20609544, 60963094, 162323448, 396155466, 899046952, 1917743448, 3879011584, 7491080844, 13892164232, 24854703014, 43071383040, 72532831794, 119038462248, 190849299076

Offset

5,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 5..1000

Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016

V. Kotesovec, Non-attacking chess pieces

Index entries for linear recurrences with constant coefficients, signature (1, -3, -7, -3, 11, 21, 13, -13, -41, -44, -8, 49, 81, 57, -15, -88, -106, -48, 48, 106, 88, 15, -57, -81, -49, 8, 44, 41, 13, -13, -21, -11, 3, 7, 3, -1, -1).

Formula

Explicit formula (Vaclav Kotesovec, Apr 04 2010): a(n) = 1/120*n^10 - 5/18*n^9 + 301/72*n^8 - 1679/45*n^7 + 78383/360*n^6 - 77519/90*n^5 + 1867681/810*n^4 - 6499681/1620*n^3 + 5324093/1296*n^2 - 12758453/6480*n + 13038851/64800 + (1/8*n^5 - 143/48*n^4 + 82/3*n^3 - 5647/48*n^2 + 10475/48*n - 3547/32)*(-1)^n + (29/2*n - 35/2)*cos(Pi*n/2) + (2*n+15)*sin(Pi*n/2) + (32/27*n^3 - 1328/81*n^2 + 6328/81*n - 5488/81)*cos(2*Pi*n/3) + (40*sqrt(3)/81*n^2 - 1496*sqrt(3)/243*n + 7024*sqrt(3)/243)*sin(2*Pi*n/3) + ((8*sqrt(5)/25 + 8/5)*n - 16*sqrt(5)/25 - 64/25)*cos(2*Pi*n/5) + 8*sqrt(22*sqrt(5)+50)/25*sin(2*Pi*n/5) + ((8/5-8*sqrt(5)/25)*n+16*sqrt(5)/25-64/25)*cos(Pi*n/5)*(-1)^n - 8*sqrt(50-22*sqrt(5))/25*sin(Pi*n/5)*(-1)^n. - Vaclav Kotesovec, Apr 04 2010

G.f.: -x^5*(14206*x^31+150238*x^30+916976*x^29+3972232*x^28+13522008*x^27+37968860*x^26+90996604*x^25+190236360*x^24+352607230*x^23+586165718*x^22+881664746*x^21+1207443842*x^20+1512654886*x^19+1738866194*x^18+1837742548*x^17+1786911600*x^16+1598078300*x^15+1312598856*x^14+987611934*x^13+677994354*x^12+422347390*x^11+236939238*x^10+118533110*x^9+52176470*x^8+19855936*x^7+6376140*x^6+1672768*x^5+341612*x^4+50540*x^3+4836*x^2+258*x+10)/((x-1)^11*(x+1)^6*(x^2+1)^2*(x^2+x+1)^4*(x^4+x^3+x^2+x+1)^2),

Recurrence: a(n)= - a(n-1) + 3*a(n-2) + 7*a(n-3) + 3*a(n-4) - 11*a(n-5) - 21*a(n-6) - 13*a(n-7) + 13*a(n-8) + 41*a(n-9) + 44*a(n-10) + 8*a(n-11) - 49*a(n-12) - 81*a(n-13) - 57*a(n-14) + 15*a(n-15) + 88*a(n-16) +106*a(n-17) + 48*a(n-18) - 48*a(n-19) -106*a(n-20) - 88*a(n-21) - 15*a(n-22) + 57*a(n-23) + 81*a(n-24) + 49*a(n-25) - 8*a(n-26) - 44*a(n-27) - 41*a(n-28) - 13*a(n-29) + 13*a(n-30) + 21*a(n-31) + 11*a(n-32) - 3*a(n-33) - 7*a(n-34) - 3*a(n-35) + a(n-36) + a(n-37). - Vaclav Kotesovec, Apr 05 2010

Mathematica

CoefficientList[Series[-(14206 x^31 + 150238*x^30 + 916976 x^29 + 3972232 x^28 + 13522008 x^27 + 37968860 x^26 + 90996604 x^25 + 190236360 x^24 + 352607230 x^23 + 586165718 x^22 + 881664746 x^21 + 1207443842 x^20 + 1512654886 x^19 + 1738866194 x^18 + 1837742548 x^17 + 1786911600 x^16 + 1598078300 x^15 + 1312598856 x^14 + 987611934 x^13 + 677994354 x^12 + 422347390 x^11 + 236939238 x^10 + 118533110 x^9 + 52176470 x^8 + 19855936 x^7 + 6376140 x^6 + 1672768 x^5 + 341612 x^4 + 50540 x^3 + 4836 x^2 + 258 x + 10) / ((x - 1)^11 (x + 1)^6 (x^2 + 1)^2 (x^2 + x + 1)^4 (x^4 + x^3 + x^2 + x + 1)^2), {x, 0, 35}], x] (* Vincenzo Librandi, May 16 2013 *)

Crossrefs

Cf. A047659, A061994.

Column k=5 of A348129.

Sequence in context: A197437 A268545 A211093 * A265854 A128013 A178021

Adjacent sequences: A108789 A108790 A108791 * A108793 A108794 A108795

Keyword

nonn,nice,easy

Author

Sergey Perepechko, Jul 09 2005

Status

approved