A076145 -id:A076145 - cp's OEIS frontend

A234363 a(n) = floor(agm(n, triangular(n))), where agm denotes the arithmetic-geometric mean.

0, 1, 2, 4, 6, 9, 12, 15, 19, 23, 27, 32, 37, 42, 48, 54, 60, 66, 73, 80, 88, 95, 103, 111, 120, 129, 138, 147, 156, 166, 176, 187, 197, 208, 219, 231, 242, 254, 267, 279, 292, 305, 318, 331, 345, 359, 373, 388, 402, 417, 432, 448, 464, 479, 496, 512, 529, 546

Offset: 0

Alex Ratushnyak, Dec 24 2013

nonn

Cf. A000217, A061979, A076145, A234362.

Table[Floor[ArithmeticGeometricMean[n,(n(n+1))/2]],{n,0,60}] (* Harvey P. Dale, Aug 17 2014 *)

a(n) ~ Pi*n^2/(4*log(2*n)). - Vaclav Kotesovec, May 09 2016

A232424 Floor of the half derivative of x^2 at n.

Original entry on oeis.org

0, 1, 4, 7, 12, 16, 22, 27, 34, 40, 47, 54, 62, 70, 78, 87, 96, 105, 114, 124, 134, 144, 155, 165, 176, 188, 199, 211, 222, 234, 247, 259, 272, 285, 298, 311, 324, 338, 352, 366, 380, 394, 409, 424, 439, 454, 469, 484, 500, 516, 531, 547, 564, 580, 597

Offset: 0

John R Phelan, Nov 23 2013

			a(4) = floor(8*4^(3/2)/(3*sqrt(Pi))) = floor(12.03604...) = 12.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Mhenni Benghorbal, Half derivative of x^2
Wikipedia, Fractional calculus

Cf. A076145, A231603.

public class Hdx2 {public static void main(String[] args) {String str = ""; for (int n = 0; str.length() < 250; n++) {long f = (long) Math.floor(8 * Math.pow(n, 1.5) / (3 * Math.sqrt(Math.PI)));str += f + ", ";} System.out.println(str);} }

a(n)=2*n^(3/2)\gamma(5/2) \\ Charles R Greathouse IV, Nov 23 2013

a(n) = floor(8*n^(3/2)/(3*sqrt(Pi))).
The d-th derivative of x^p is p!*x^(p-d)/(p-d)!, as long as (p-d) is not a negative integer.
For p = 2, d = 1/2 2!x^(3/2)/(3/2)! = 2x^(3/2)/((3/2)*(1/2)!) = 2x^(3/2)/((3/2)*sqrt(Pi)/2) = 8x^(3/2)/(3*sqrt(Pi)).
Note that 1.5! = Gamma(5/2).

A234362 Floor(AGM(n^2, n^3)), where AGM denotes the arithmetic-geometric mean.

Original entry on oeis.org

0, 1, 5, 16, 35, 65, 106, 161, 231, 318, 425, 551, 700, 872, 1069, 1293, 1545, 1827, 2140, 2486, 2866, 3281, 3734, 4225, 4755, 5327, 5942, 6601, 7306, 8057, 8856, 9706, 10606, 11558, 12565, 13626, 14744, 15919, 17154, 18449, 19805, 21225, 22709, 24259, 25876, 27561

Offset: 0

Alex Ratushnyak, Dec 24 2013

nonn

Cf. A000290, A000578, A061979, A076145, A234363.

import math
for n in range(67):
  x = n*n
  y = x*n
  a = (x+y)*0.5
  g = math.sqrt(x*y)
  while abs(a-g)>0.00000000001:
    x = a
    y = g
    a = (x+y)*0.5
    g = math.sqrt(x*y)
  print(int(a), end=', ')

cp's OEIS Frontend

A234363 a(n) = floor(agm(n, triangular(n))), where agm denotes the arithmetic-geometric mean.

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A232424 Floor of the half derivative of x^2 at n.

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A234362 Floor(AGM(n^2, n^3)), where AGM denotes the arithmetic-geometric mean.

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