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A232424
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Floor of the half derivative of x^2 at n.
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1
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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
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = floor(8n^(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).
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EXAMPLE
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For n = 4, a(n) = floor(8n^(3/2)/(3*sqrt(Pi)) = floor(8*4^(3/2)/(3*1.77245...) = floor(8*8/5.31736...) = floor(12.03604...) = 12.
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PROG
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(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); } }
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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