0,5

An analytical solution for the 4-queens problem permits us to combine six particular cases into a single "unified" expression: a(n)=n(n-1)(45n^6-855n^5+6945n^4-30891n^3+78864n^2-106226n+53404)/1080 + (n^3-21/2n^2+28n-14)[n/2]+32/9(n-1)[n/3]+(16/9n-4)[(n+1)/3], where [x]=floor(x). The method used to derive this formula helps to fine-tune an estimate by E. Lucas for a(n) (see comment to A047659 "3-queens problem"). For any fixed value of k>1, a(n)=n^(2k)/k!-5/3n^(2k-1)/(k-2)!+O(n^(2k-2)) - Sergey Perepechko (persn(AT)aport.ru), Sep 16 2005

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.

G.f.: - (574*x^16 + 3804*x^15 + 13522*x^14 + 29768*x^13 + 2*x^4 + 46890*x^12 + 76*x^5 + 53580*x^11 + 734*x^6 + 46304*x^10 + 3992*x^7 + 29356*x^9 + 13318*x^8)/( - 1 + x^17 - 3*x^16 - x^15 + 9*x^14 - 12*x^12 - 7*x^11 + 15*x^10 + 16*x^9 - 16*x^8 - 15*x^7 + 7*x^6 + 12*x^5 - 9*x^3 + x^2 + 3*x).

Recurrence: a(n) = 3*a(n - 1) + a(n - 2) - 9*a(n - 3) + 12*a(n - 5) + 7*a(n - 6) - 15*a(n - 7) - 16*a(n - 8) + 16*a(n - 9) + 15*a(n - 10) - 7*a(n - 11) - 12*a(n - 12) + 9*a(n - 14) - a(n - 15) - 3*a(n - 16) + a(n - 17), n >= 17.

Explicit formula (V. Kotesovec, 1992) for n >= 2: a(n) = n^8/24 - 5*n^7/6 + 65*n^6/9 - 1051*n^5/30 + 817*n^4/8 added to one of the following terms:

- 4769*n^3/27 + 1963*n^2/12 - 1769*n/30 if n = 0 (mod 6) - 9565*n^3/54 + 1013*n^2/6 - 6727*n/90 + 257/27 if n = 1 (mod 6) - 4769*n^3/27 + 1963*n^2/12 - 5467*n/90 + 28/27 if n = 2 (mod 6) - 9565*n^3/54 + 1013*n^2/6 - 2189*n/30 + 7 if n = 3 (mod 6) - 4769*n^3/27 + 1963*n^2/12 - 5467*n/90 + 68/27 if n = 4 (mod 6) - 9565*n^3/54 + 1013*n^2/6 - 6727*n/90 + 217/27 if n = 5 (mod 6)

Cf. A036464, A047659.

Cf. A047659.

Sequence in context: A090603 A020955 A120826 this_sequence A093666 A063270 A087617

Adjacent sequences: A061991 A061992 A061993 this_sequence A061995 A061996 A061997

nonn

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