Ugly But Interesting Harmonic Number Sequence

Leroy Quet qq-quet at mindspring.com
Sat Oct 21 22:07:52 CEST 2006


>Let H(n) = sum{k=1 to n} 1/k, the nth harmonic number.
>(H(0) = 0.)
>
>Then, for n = any *positive* integer,
>
>a(n) = ( (2n)!*(1 -H(2n))/2 + (2n+1)!*(H(n+1)/(n+2) - H(2n+2)/(2n+3)) 
>)/(n+1)
>
>is always an integer.

Of course, the above can be somewhat simplified (since (2n)!/(2(n+1)) is 
an integer for n >= 2):


For n >= 2,

b(n) = (2n+1)!*( H(n+1)/(n+2) - H(2n)/(2(2n+1)) - H(2n+2)/(2n+3) )/(n+1)

is always an integer.


>
>I based this result on an identity I found years ago which I cannot 
>remember how I got it. So perhaps the above result is wrong.
>
>But my question to seq.fan is, is it appropriate to submit unaestheticly 
>generated integer sequences which are interesting solely because every 
>term is an integer?
>
>And if so, and if each a(n) is an integer, could someone please calculate 
>and submit the sequence {a(k)}?
>
>Thanks,
>Leroy Quet






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