While perusing Wolfram Alpha, I stumbled upon a strange curiosity involving the Gamma function:

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> The *arc length* of $\Gamma(x + 1)$ from 1 to $x$ *very* closely equals the numeric value of $\Gamma(x)$.
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Examples for $x = $...

1. (Not evaluated)
2. $\Gamma(2+1) = 2;~~~~~~$ Arc length of $\Gamma(x+1)$ from 1 to 2 = [1.442...](http://www.wolframalpha.com/input/?i=arc+length+of+gamma(x%2B1)+from+1+to+2)
3. $\Gamma(3+1) = 6;~~~~~~$ Arc length of $\Gamma(x+1)$ from 1 to 3 = [5.5846...](http://www.wolframalpha.com/input/?i=arc+length+of+gamma(x%2B1)+from+1+to+3)
4. $\Gamma(4+1) = 24;~~~~$ Arc length of $\Gamma(x+1)$ from 1 to 4 = [23.6185...](http://www.wolframalpha.com/input/?i=arc+length+of+gamma(x%2B1)+from+1+to+4)
5. $\Gamma(5+1) = 120;~~$ Arc length of $\Gamma(x+1)$ from 1 to 5 = [119.625...](http://www.wolframalpha.com/input/?i=arc+length+of+gamma(x%2B1)+from+1+to+5)
6. $\Gamma(6+1) = 720;~~$ Arc length of $\Gamma(x+1)$ from 1 to 6 = [719.626...](http://www.wolframalpha.com/input/?i=arc+length+of+gamma(x%2B1)+from+1+to+6)

Question
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> **Is this an unusual or seldom-discussed-but-fascinating mathematical coincidence?**
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Or, am I missing some obvious numerical relationship staring me right in my eyes? ;-)

P.S. I *am* aware of how the gamma function extends factorial to $\mathbb{R}$, but with the $+1$ offset.