A231603 Floor of the arithmetic-geometric mean of n+n and n*n.

0, 1, 4, 7, 11, 16, 22, 28, 35, 43, 52, 61, 70, 81, 92, 103, 115, 128, 141, 155, 170, 185, 200, 216, 233, 250, 268, 286, 305, 325, 344, 365, 386, 408, 430, 452, 475, 499, 523, 548, 573, 598, 625, 651, 678, 706, 734, 763, 792, 822, 852, 883, 914, 945, 978, 1010, 1043, 1077, 1111, 1145, 1180, 1216, 1252, 1288, 1325

Offset

0,3

Comments

Arithmetic-geometric mean of n+n and n*n.

AGM of the sum and product of n and n.

a(n) = agm(A005843(n), A000290(n)).

Links

Table of n, a(n) for n=0..64.

Wikipedia, Arithmetic-geometric mean

Formula

a(n) = floor(agm(n+n,n*n)).

Example

a(2) = floor(agm(2+2, 2*2)) = floor(agm(4, 4)) = 4.
a(5) = floor(agm(10.0, 25.0)) = floor(agm(17.5, 15.811388)) = floor(agm(16.655695, 16.634281)) = floor(agm(16.644987, 16.644983)) = floor(16.644987) = 16.

Mathematica

Table[Floor[ArithmeticGeometricMean[2n, n^2]], {n, 0, 70}] (* Harvey P. Dale, Aug 03 2014 *)

Prog

(Java) public class Agmnxn {
    private static final double TOLERANCE = Math.pow(10, -4);
    private static final long LENGTH = 250;
    public static void main(String[] args) {
       String series="";
       long n=0;
       while (series.length()<LENGTH) {
          long x=(long) Math.floor(agm(n+n, n*n));
          series+=x+", ";
          n++;
       }
       System.out.println(series);
    }
    private static double agm(double a, double g) {
      return Math.abs(a-g)<TOLERANCE?a:agm((a+g)/2, Math.sqrt(a*g));
    }
}

Crossrefs

Cf. A005843, A005843, A090852.

Sequence in context: A057054 A269132 A310757 * A134869 A171377 A020683

Adjacent sequences: A231600 A231601 A231602 * A231604 A231605 A231606

Keyword

nonn,easy

Author

John R Phelan, Nov 11 2013

Status

approved