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

%I A231603 #16 Jun 20 2021 19:18:13
%S A231603 0,1,4,7,11,16,22,28,35,43,52,61,70,81,92,103,115,128,141,155,170,185,
%T A231603 200,216,233,250,268,286,305,325,344,365,386,408,430,452,475,499,523,
%U A231603 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
%N A231603 Floor of the arithmetic-geometric mean of n+n and n*n.
%C A231603 Arithmetic-geometric mean of n+n and n*n.
%C A231603 AGM of the sum and product of n and n.
%C A231603 a(n) = agm(A005843(n), A000290(n)).
%H A231603 Wikipedia, <a href="http://en.wikipedia.org/wiki/Arithmetic_geometric_mean">Arithmetic-geometric mean</a>
%F A231603 a(n) = floor(agm(n+n,n*n)).
%e A231603 a(2) = floor(agm(2+2, 2*2)) = floor(agm(4, 4)) = 4.
%e A231603 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.
%t A231603 Table[Floor[ArithmeticGeometricMean[2n,n^2]],{n,0,70}] (* _Harvey P. Dale_, Aug 03 2014 *)
%o A231603 (Java) public class Agmnxn {
%o A231603     private static final double TOLERANCE = Math.pow(10, -4);
%o A231603     private static final long LENGTH = 250;
%o A231603     public static void main(String[] args) {
%o A231603        String series="";
%o A231603        long n=0;
%o A231603        while (series.length()<LENGTH) {
%o A231603           long x=(long) Math.floor(agm(n+n,n*n));
%o A231603           series+=x+",";
%o A231603           n++;
%o A231603        }
%o A231603        System.out.println(series);
%o A231603     }
%o A231603     private static double agm(double a, double g) {
%o A231603       return Math.abs(a-g)<TOLERANCE?a:agm((a+g)/2,Math.sqrt(a*g));
%o A231603     }
%o A231603 }
%Y A231603 Cf. A005843, A005843, A090852.
%K A231603 nonn,easy
%O A231603 0,3
%A A231603 _John R Phelan_, Nov 11 2013