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A201629 a(n) = n if n is even and otherwise its nearest multiple of 4.

%I A201629 #60 Sep 08 2022 08:46:01
%S A201629 0,0,2,4,4,4,6,8,8,8,10,12,12,12,14,16,16,16,18,20,20,20,22,24,24,24,
%T A201629 26,28,28,28,30,32,32,32,34,36,36,36,38,40,40,40,42,44,44,44,46,48,48,
%U A201629 48,50,52,52,52,54,56,56,56,58,60,60,60,62,64,64,64,66,68,68
%N A201629 a(n) = n if n is even and otherwise its nearest multiple of 4.
%C A201629 For n > 1, the maximal number of nonattacking knights on a 2 x (n-1) chessboard.
%C A201629 Compare this with the binary triangle construction of A240828.
%C A201629 Minimal number of straight segments in a rook circuit of an (n-1) X n board (see example). - _Ruediger Jehn_, Feb 26 2021
%H A201629 Vincenzo Librandi, <a href="/A201629/b201629.txt">Table of n, a(n) for n = 0..1000</a>
%H A201629 Rüdiger Jehn, <a href="https://arxiv.org/abs/2103.15778">Properties of Hamiltonian Circuits in Rectangular Grids</a>, arXiv:2103.15778 [math.GM], 2021.
%H A201629 Craig Knecht, <a href="/A201629/a201629.jpg">Row sums of superimposed and added binary filled triangles.</a>
%H A201629 Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>
%F A201629 a(n) = n - sin(n*Pi/2).
%F A201629 G.f.: 2*x^2/((1-x)^2*(1+x^2)).
%F A201629 a(n) = 2*A004524(n+1). - _R. J. Mathar_, Feb 02 2012
%F A201629 a(n) = n+(1-(-1)^n)*(-1)^((n+1)/2)/2. - _Bruno Berselli_, Aug 06 2014
%F A201629 E.g.f.: x*exp(x) - sin(x). - _G. C. Greubel_, Aug 13 2018
%e A201629 G.f. = 2*x^2 + 4*x^3 + 4*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 8*x^8 + 8*x^9 + ...
%e A201629 From _Ruediger Jehn_, Feb 26 2021: (Start)
%e A201629 a(5) = 4:
%e A201629    +----+----+----+----+----+
%e A201629    |  __|____|_   |   _|__  |
%e A201629    | /  |    | \  |  / |  \ |
%e A201629    +----+----+----+----+----+
%e A201629    | \__|__  | |  |  | |  | |
%e A201629    |    |  \ | \__|__/ |  | |
%e A201629    +----+----+----+----+----+
%e A201629    |  __|__/ |  __|__  |  | |
%e A201629    | /  |    | /  |  \ |  | |
%e A201629    +----+----+----+----+----+
%e A201629    | \  |    | |  |  | |  | |
%e A201629    |  \_|____|_/  |  \_|__/ |
%e A201629    +----+----+----+----+----+
%e A201629 There are at least 4 squares on the 4 X 5 board with straight lines (here in squares a_12, a_25, a_35 and a_42).  (End)
%p A201629 seq(n-sin(Pi*n/2), n=0..30); # _Robert Israel_, Jul 14 2015
%t A201629 Table[2*(Floor[(Floor[(n + 1)/2] + 1)/2] + Floor[(Floor[n/2] + 1)/2]), {n, 1, 100}]
%t A201629 Table[If[EvenQ[n], n, 4*Round[n/4]], {n, 0, 68}] (* _Alonso del Arte_, Jan 27 2012 *)
%t A201629 CoefficientList[Series[2 x^2/((-1 + x)^2 (1 + x^2)), {x, 0, 100}], x] (* _Vincenzo Librandi_, Aug 06 2014 *)
%t A201629 a[ n_] := n - KroneckerSymbol[ -4, n]; (* _Michael Somos_, Jul 18 2015 *)
%o A201629 (PARI) a(n)=n\4*4+[0, 0, 2, 4][n%4+1] \\ _Charles R Greathouse IV_, Jan 27 2012
%o A201629 (PARI) {a(n) = n - kronecker( -4, n)}; /* _Michael Somos_, Jul 18 2015 */
%o A201629 (Haskell)
%o A201629 a201629 = (* 2) . a004524 . (+ 1) -- _Reinhard Zumkeller_, Aug 05 2014
%o A201629 (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(2*x^2/((1-x)^2*(1+x^2)))); // _G. C. Greubel_, Aug 13 2018
%Y A201629 Cf. A004524, A085801, A189889, A190394.
%K A201629 nonn,easy
%O A201629 0,3
%A A201629 _Vaclav Kotesovec_, Dec 03 2011
%E A201629 Formula corrected by _Robert Israel_, Jul 14 2015