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 A030978 Maximal number of non-attacking knights on an n X n board. 11
 0, 1, 4, 5, 8, 13, 18, 25, 32, 41, 50, 61, 72, 85, 98, 113, 128, 145, 162, 181, 200, 221, 242, 265, 288, 313, 338, 365, 392, 421, 450, 481, 512, 545, 578, 613, 648, 685, 722, 761, 800, 841, 882, 925, 968, 1013, 1058, 1105, 1152, 1201, 1250, 1301, 1352, 1405 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS In other words, independence number of the n X n knight graph. - Eric W. Weisstein, May 05 2017 REFERENCES H. E. Dudeney, The Knight-Guards, #319 in Amusements in Mathematics; New York: Dover, p. 95, 1970. J. S. Madachy, Madachy's Mathematical Recreations, New York, Dover, pp. 38-39 1979. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 751. Eric Weisstein's World of Mathematics, Independence Number Eric Weisstein's World of Mathematics, Knight Graph Eric Weisstein's World of Mathematics, Knights Problem Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1). FORMULA a(n) = 4 if n = 2, n^2/2 if n even > 2, (n^2+1)/2 if n odd > 1. a(n) = 4 if n = 2, (1 + (-1)^(1 + n) + 2 n^2)/4 otherwise. G.f.: x*(2*x^5-4*x^4+3*x^2-2*x-1) / ((x-1)^3*(x+1)). [Colin Barker, Jan 09 2013] MATHEMATICA CoefficientList[Series[x (2 x^5 - 4 x^4 + 3 x^2 - 2 x - 1)/((x - 1)^3 (x + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 19 2013 *) Join[{0, 1, 4}, Table[If[EvenQ[n], n^2/2, (n^2 + 1)/2], {n, 3, 60}]] (* Harvey P. Dale, Nov 28 2014 *) Join[{0, 1, 4}, LinearRecurrence[{2, 0, -2, 1}, {5, 8, 13, 18}, 60]] (* Harvey P. Dale, Nov 28 2014 *) Table[If[n == 2, 4, (1 - (-1)^n + 2 n^2)/4], {n, 20}] (* Eric W. Weisstein, May 05 2017 *) Table[Length[FindIndependentVertexSet[KnightTourGraph[n, n]][[1]]], {n, 20}] (* Eric W. Weisstein, Jun 27 2017 *) CROSSREFS Agrees with A000982 for n>1. Cf. A244081. Sequence in context: A133940 A174398 A341420 * A101948 A348484 A087475 Adjacent sequences: A030975 A030976 A030977 * A030979 A030980 A030981 KEYWORD nonn,easy AUTHOR EXTENSIONS More terms from Erich Friedman Definition clarified by Vaclav Kotesovec, Sep 16 2014 STATUS approved

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Last modified February 5 03:17 EST 2023. Contains 360082 sequences. (Running on oeis4.)