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

0,3

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

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

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

Example

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)

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 *)

Prog

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

Crossrefs

Cf. A004524, A085801, A189889, A190394.

Sequence in context: A235790 A023988 A023819 * A033827 A008217 A170889

Adjacent sequences: A201626 A201627 A201628 * A201630 A201631 A201632

Keyword

nonn,easy

Author

Vaclav Kotesovec, Dec 03 2011

Extensions

Formula corrected by Robert Israel, Jul 14 2015

Status

approved