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

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

Author

John R Phelan, Nov 23 2013

Keywords

• nonn
• easy

Examples

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

Links

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

Crossrefs

Cf. A076145, A231603.

Programs

Java

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);} }

PARI

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

Formula

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).