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A189889 Maximum number of nonattacking kings on an n X n toroidal board. 8
1, 1, 1, 4, 5, 9, 10, 16, 18, 25, 27, 36, 39, 49, 52, 64, 68, 81, 85, 100, 105, 121, 126, 144, 150, 169, 175, 196, 203, 225, 232, 256, 264, 289, 297, 324, 333, 361, 370, 400, 410, 441, 451, 484, 495, 529, 540, 576, 588, 625 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

REFERENCES

John Watkins, Across the Board: The Mathematics of Chessboard Problems (2004), Theorem 11.1, p.194.

LINKS

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

Hernan de Alba, W. Carballosa, J. LeaƱos, L. M. Rivera, Independence and matching numbers of some token graphs, arXiv preprint arXiv:1606.06370 [math.CO], 2016.

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 751.

E. Weisstein, Kings Problem, MathWorld

Index entries for linear recurrences with constant coefficients, signature (1, 1, -1, 1, -1, -1, 1).

FORMULA

Explicit formula (Watkins and Ricci, 2004): a(n) = floor((n*floor(n/2))/2), n > 1.

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7).

G.f.: x*(-x^7 +x^6 +x^5 +3*x^3 -x^2 +1) / (-x^7 +x^6 +x^5 -x^4+ x^3 -x^2 -x +1).

MAPLE

A189889:=n->`if`(n=1, 1, floor(n*floor(n/2)/2)); seq(A189889(k), k=1..100); # Wesley Ivan Hurt, Nov 07 2013

MATHEMATICA

Table[If[n==1, 1, Floor[(n*Floor[n/2])/2]], {n, 1, 50}]

CoefficientList[Series[(- x^7 + x^6 + x^5 + 3 * x^3 - x^2 + 1) / (-x^7 + x^6 + x^5 - x^4 + x^3 - x^2 - x + 1), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 02 2013 *)

Join[{1}, LinearRecurrence[{1, 1, -1, 1, -1, -1, 1}, {1, 1, 4, 5, 9, 10, 16}, 50]] (* Harvey P. Dale, Aug 07 2013 *)

PROG

(PARI) print(Vec(x*(-x^7 + x^6 + x^5 + 3*x^3 - x^2 + 1) / (-x^7 + x^6 + x^5 - x^4 + x^3 - x^2 - x + 1) + O(x^51))); /* or */

for(n=1, 50, print1(if(n==1, 1, floor((n*floor(n/2))/2)), ", ")); \\ Indranil Ghosh, Mar 09 2017

(Python) def A189889(n): return 1 if n==1 else (n*(n/2))/2 # Indranil Ghosh, Mar 09 2017

(MAGMA) [1] cat [Floor(n*Floor(n/2)/2): n in [2..50]]; // G. C. Greubel, Jan 13 2018

CROSSREFS

Cf. A018807, A085801, A172158, A174558, A179428, A180067.

Sequence in context: A050036 A059582 A257058 * A034809 A096808 A067261

Adjacent sequences:  A189886 A189887 A189888 * A189890 A189891 A189892

KEYWORD

nonn,easy

AUTHOR

Vaclav Kotesovec, Apr 30 2011

STATUS

approved

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Last modified May 21 15:32 EDT 2019. Contains 323444 sequences. (Running on oeis4.)