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A189889 Maximum number of nonattacking kings on an n X n toroidal board. 9
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
Hernan de Alba, W. Carballosa, J. Leaños, 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
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
Sequence in context: A050036 A059582 A257058 * A329451 A308578 A034809
KEYWORD
nonn,easy
AUTHOR
Vaclav Kotesovec, Apr 30 2011
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)