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

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, 26, 28, 28, 28, 30, 32, 32, 32, 34, 36, 36, 36, 38, 40, 40, 40, 42, 44, 44, 44, 46, 48, 48, 48, 50, 52, 52, 52, 54, 56, 56, 56, 58, 60, 60, 60, 62, 64, 64, 64, 66, 68, 68

Offset: 0

Author

Vaclav Kotesovec, Dec 03 2011

Keywords

• nonn
• easy

Comments

For n > 1, the maximal number of nonattacking knights on a 2 x (n-1) chessboard.

Compare this with the binary triangle construction of A240828.

Minimal number of straight segments in a rook circuit of an (n-1) X n board (see example). - Ruediger Jehn, Feb 26 2021

Examples

			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 + ...
From _Ruediger Jehn_, Feb 26 2021: (Start)
a(5) = 4:
   +----+----+----+----+----+
   |  __|____|_   |   _|__  |
   | /  |    | \  |  / |  \ |
   +----+----+----+----+----+
   | \__|__  | |  |  | |  | |
   |    |  \ | \__|__/ |  | |
   +----+----+----+----+----+
   |  __|__/ |  __|__  |  | |
   | /  |    | /  |  \ |  | |
   +----+----+----+----+----+
   | \  |    | |  |  | |  | |
   |  \_|____|_/  |  \_|__/ |
   +----+----+----+----+----+
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)
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Rüdiger Jehn, Properties of Hamiltonian Circuits in Rectangular Grids, arXiv:2103.15778 [math.GM], 2021.
• Craig Knecht, Row sums of superimposed and added binary filled triangles.
• Vaclav Kotesovec, Non-attacking chess pieces

Crossrefs

Cf. A004524, A085801, A189889, A190394.

Programs

Haskell

a201629 = (* 2) . a004524 . (+ 1) -- Reinhard Zumkeller, Aug 05 2014

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(2*x^2/((1-x)^2*(1+x^2)))); // G. C. Greubel, Aug 13 2018

Maple

seq(n-sin(Pi*n/2), n=0..30); # Robert Israel, Jul 14 2015

Mathematica

Table[2*(Floor[(Floor[(n + 1)/2] + 1)/2] + Floor[(Floor[n/2] + 1)/2]), {n, 1, 100}]
Table[If[EvenQ[n], n, 4*Round[n/4]], {n, 0, 68}] (* Alonso del Arte, Jan 27 2012 *)
CoefficientList[Series[2 x^2/((-1 + x)^2 (1 + x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 06 2014 *)
a[ n_] := n - KroneckerSymbol[ -4, n]; (* Michael Somos, Jul 18 2015 *)

PARI

a(n)=n\4*4+[0, 0, 2, 4][n%4+1] \\ Charles R Greathouse IV, Jan 27 2012

PARI

{a(n) = n - kronecker( -4, n)}; /* Michael Somos, Jul 18 2015 */

Formula

a(n) = n - sin(n*Pi/2).

G.f.: 2*x^2/((1-x)^2*(1+x^2)).

a(n) = 2*A004524(n+1). - R. J. Mathar, Feb 02 2012

a(n) = n+(1-(-1)^n)*(-1)^((n+1)/2)/2. - Bruno Berselli, Aug 06 2014

E.g.f.: x*exp(x) - sin(x). - G. C. Greubel, Aug 13 2018

Extensions

Formula corrected by Robert Israel, Jul 14 2015