login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A061994 Number of ways to place 4 nonattacking queens on an n X n board. 16
0, 0, 0, 0, 2, 82, 982, 7002, 34568, 131248, 412596, 1123832, 2739386, 6106214, 12654614, 24675650, 45704724, 80999104, 138170148, 227938788, 365106738, 569681574, 868289594, 1295775946, 1897176508, 2729909796 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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, Sep 16 2005

REFERENCES

Vaclav Kotesovec, Between chessboard and computer, 1996, pp. 204-206.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Louis Azemard, Une communication de Vaclav Kotesovec, Echecs et Mathématiques, Rex Multiplex 38/1992.

Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 12.

Index entries for linear recurrences with constant coefficients, signature (3, 1, -9, 0, 12, 7, -15, -16, 16, 15, -7, -12, 0, 9, -1, -3, 1).

FORMULA

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).

a(n) = n^8/24 - 5n^7/6 + 65n^6/9 - 1051n^5/30 + 817n^4/8 - 19103n^3/108 + 3989n^2/24 - 18131n/270 + 253/54 + (n^3/4 - 21n^2/8 + 7n - 7/2)*(-1)^n + 32*(n - 1)/27*cos(2*Pi*n/3) + 40/81*sqrt(3)*sin(2*Pi*n/3). - Vaclav Kotesovec, Feb 11 2010

E.g.f.: (3*(exp(2*x)*(5060 - 4645*x + 1755*x^2 - 590*x^3 + 480*x^4 + 414*x^5 + 870*x^6 + 360*x^7 + 45*x^8) - 135*(28 + 37*x + 15*x^2 + 2*x^3)) - 1920 * exp(x/2) * (2+x) * cos(sqrt(3)*x/2) - 320 * sqrt(3) * exp(x/2) * (6*x-5) * sin(sqrt(3)*x/2)) / (3240 * exp(x)). - Vaclav Kotesovec, Feb 15 2015

MATHEMATICA

CoefficientList[Series[-(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), {x, 0, 40}], x] (* Vincenzo Librandi, May 12 2013 *)

LinearRecurrence[{3, 1, -9, 0, 12, 7, -15, -16, 16, 15, -7, -12, 0, 9, -1, -3, 1}, {0, 0, 0, 0, 2, 82, 982, 7002, 34568, 131248, 412596, 1123832, 2739386, 6106214, 12654614, 24675650, 45704724}, 40] (* Harvey P. Dale, Jan 21 2017 *)

CROSSREFS

Cf. A036464, A047659.

Sequence in context: A120826 A259308 A202965 * A197641 A093666 A246002

Adjacent sequences:  A061991 A061992 A061993 * A061995 A061996 A061997

KEYWORD

nonn,nice,easy

AUTHOR

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

EXTENSIONS

Minor edits by Vaclav Kotesovec, Feb 15 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified June 23 15:27 EDT 2017. Contains 288666 sequences.