A172135 Number of ways to place 4 nonattacking knights on an n X n board.

0, 1, 18, 412, 4436, 26133, 111066, 376560, 1080942, 2732909, 6253408, 13204356, 26100160, 48819677, 87137934, 149398608, 247349946, 397168485, 620696612, 946921684, 1413726108, 2069939461, 2977725410, 4215337872

Offset

1,3

References

E. Bonsdorff, K. Fabel, O. Riihimaa, Schach und Zahl, 1966, p. 51-63

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Formula

a(n) = (n^8 - 54*n^6 + 144*n^5 + 1019*n^4 - 5232*n^3 - 2022*n^2 + 51120*n - 77184)/24, n >= 6. (Karl Fabel, 1966)

G.f.: x^2 * ( 1 + 9*x + 286*x^2 + 1292*x^3 - 345*x^4 +3099*x^5 - 5142*x^6 + 3606*x^7 - 1162*x^8 - 390*x^9 + 690*x^10 - 312*x^11 + 48*x^12) / (1-x)^9. - Vaclav Kotesovec, Mar 25 2010

E.g.f.: x^2/2! + 18*x^3/3! + 412*x^4/4! + 4436*x^5/5! + (1/120)*(385920 + 161040*x + 17940*x^2 - 1200*x^3 - 2660*x^4 - 4484*x^5 + (-385920 + 224880*x - 49860*x^2 + 2940*x^3 + 3250*x^4 + 1920*x^5 + 1060*x^6 + 140*x^7 + 5*x^8)*exp(x)). - G. C. Greubel, Apr 19 2022

Mathematica

CoefficientList[Series[x*(1 +9*x +286*x^2 +1292*x^3 -345*x^4 +3099*x^5 -5142*x^6 +3606*x^7 -1162*x^8 -390*x^9 +690*x^10 -312*x^11 +48*x^12)/(1-x)^9, {x, 0, 40}], x] (* Vincenzo Librandi, May 26 2013 *)

Prog

(Magma) [0, 1, 18, 412, 4436] cat [(n^8 -54*n^6 +144*n^5 +1019*n^4 -5232*n^3 -2022*n^2 +51120*n -77184)/24: n in [6..50]]; // G. C. Greubel, Apr 19 2022
(SageMath) [0, 1, 18, 412, 4436] + [(n^8 -54*n^6 +144*n^5 +1019*n^4 -5232*n^3 -2022*n^2 +51120*n -77184)/24 for n in (6..50)] # G. C. Greubel, Apr 19 2022

Crossrefs

Cf. A061994, A172127, A172132, A172134.

Column k=4 of A244081.

Sequence in context: A260655 A318598 A215229 * A005477 A361549 A326368

Adjacent sequences: A172132 A172133 A172134 * A172136 A172137 A172138

Keyword

nonn,easy

Author

Vaclav Kotesovec, Jan 26 2010

Status

approved