Cut the Cards
08-25-2020, 06:14 PM (This post was last modified: 08-28-2020 01:23 AM by Albert Chan.)
Post: #9
 Albert Chan Senior Member Posts: 1,848 Joined: Jul 2018
RE: Cut the Cards
(07-30-2020 09:49 PM)Jim Horn Wrote:  So, it approaches x*(ln(x)+0.57721), where the latter constant is the Euler-Mascheroni constant.

Replacing ln(x) by ln(x+0.5) gives much better estimate for ψ(x+1).
ψ(x+1) = slope of ln(gamma(t)) at t = x+1. When x is large:

$$\psi(x+1) ≈ {\ln\Gamma(x+2)\;-\;\ln\Gamma(x)\over 2}=\ln\sqrt{(x+1)(x)} ≈ \ln(x+0.5)﻿$$

>>> x = 52
>>> H = log(x+0.5) + 0.57722
>>> print format(x*H, 'g')
235.978

see https://en.wikipedia.org/wiki/Digamma_fu...roximation
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 Messages In This Thread Cut the Cards - David Hayden - 07-30-2020, 08:00 PM RE: Cut the Cards - Albert Chan - 07-30-2020, 08:58 PM RE: Cut the Cards - Albert Chan - 08-21-2020, 11:00 PM RE: Cut the Cards - Jim Horn - 07-30-2020, 09:49 PM RE: Cut the Cards - John Keith - 07-31-2020, 12:24 AM RE: Cut the Cards - Gerson W. Barbosa - 08-24-2020, 01:57 PM RE: Cut the Cards - Albert Chan - 08-25-2020 06:14 PM RE: Cut the Cards - Albert Chan - 07-30-2020, 10:21 PM RE: Cut the Cards - pinkman - 08-24-2020, 09:49 PM RE: Cut the Cards - Gerson W. Barbosa - 08-25-2020, 11:41 PM RE: Cut the Cards - Albert Chan - 08-26-2020, 03:06 AM RE: Cut the Cards - Gerson W. Barbosa - 08-26-2020, 08:23 AM RE: Cut the Cards - Albert Chan - 08-26-2020, 02:13 PM RE: Cut the Cards - Gerson W. Barbosa - 08-26-2020, 06:13 PM RE: Cut the Cards - Gerson W. Barbosa - 08-27-2020, 10:07 PM RE: Cut the Cards - Albert Chan - 08-28-2020, 09:26 PM RE: Cut the Cards - Albert Chan - 08-29-2020, 04:02 PM RE: Cut the Cards - Gerson W. Barbosa - 08-28-2020, 11:39 PM RE: Cut the Cards - Albert Chan - 06-23-2021, 12:08 AM

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