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