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

%I A232424 #37 Nov 04 2024 02:41:59
%S A232424 0,1,4,7,12,16,22,27,34,40,47,54,62,70,78,87,96,105,114,124,134,144,
%T A232424 155,165,176,188,199,211,222,234,247,259,272,285,298,311,324,338,352,
%U A232424 366,380,394,409,424,439,454,469,484,500,516,531,547,564,580,597
%N A232424 Floor of the half derivative of x^2 at n.
%H A232424 Charles R Greathouse IV, <a href="/A232424/b232424.txt">Table of n, a(n) for n = 0..10000</a>
%H A232424 Mhenni Benghorbal, <a href="http://math.stackexchange.com/questions/198901/half-derivative-of-x2">Half derivative of x^2</a>
%H A232424 Wikipedia, <a href="http://en.wikipedia.org/wiki/Fractional_calculus">Fractional calculus</a>
%F A232424 a(n) = floor(8*n^(3/2)/(3*sqrt(Pi))).
%F A232424 The d-th derivative of x^p is p!*x^(p-d)/(p-d)!, as long as (p-d) is not a negative integer.
%F A232424 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)).
%F A232424 Note that 1.5! = Gamma(5/2).
%e A232424 a(4) = floor(8*4^(3/2)/(3*sqrt(Pi))) = floor(12.03604...) = 12.
%o A232424 (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);} }
%o A232424 (PARI) a(n)=2*n^(3/2)\gamma(5/2) \\ _Charles R Greathouse IV_, Nov 23 2013
%Y A232424 Cf. A076145, A231603.
%K A232424 nonn,easy
%O A232424 0,3
%A A232424 _John R Phelan_, Nov 23 2013