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From: Richard L. Peterson on 30 Sep 2009 21:17
1-243+3125-...(+-)(2n-1)^5=(+-)nR^2, where
R=4n^2-5, is an easy to prove identity once
you notice it. Is it already known?
From: Richard L. Peterson on 30 Sep 2009 21:19
I mean is the identity already known, not the
proof, which is easy.
From: Rick Decker on 1 Oct 2009 11:06
Richard L. Peterson wrote:
> I mean is the identity already known, not the
> proof, which is easy.

Your sequence, 1^5 - 3^5 + 5^5 - 7^5 + ...,
is not in the online encyclopedia of integer sequences.

However, a generalization of your result, for the alternating
sum of powers of odd integers, is indeed known. Search
for "Swiss-knife function". It's quite interesting.


Regards,

Rick
From: Phil Carmody on 1 Oct 2009 15:57
Rick Decker <rdecker(a)hamilton.edu> writes:

> Richard L. Peterson wrote:
>> I mean is the identity already known, not the
>> proof, which is easy.
>
> Your sequence, 1^5 - 3^5 + 5^5 - 7^5 + ...,
> is not in the online encyclopedia of integer sequences.
>
> However, a generalization of your result, for the alternating
> sum of powers of odd integers, is indeed known. Search
> for "Swiss-knife function". It's quite interesting.

If you mean "The Canonical Generating Function or CGF(z) -- a Swiss-knife
function", then it looks a load of codswallop. Then again quoting Chaitin
right at the start was bound to prejudice me against it. I gave up before
I got to anything lucid.

Phil
--
Any true emperor never needs to wear clothes. -- Devany on r.a.s.f1
From: Rick Decker on 1 Oct 2009 16:22
Phil Carmody wrote:
> Rick Decker <rdecker(a)hamilton.edu> writes:
>
>> Richard L. Peterson wrote:
>>> I mean is the identity already known, not the
>>> proof, which is easy.
>> Your sequence, 1^5 - 3^5 + 5^5 - 7^5 + ...,
>> is not in the online encyclopedia of integer sequences.
>>
>> However, a generalization of your result, for the alternating
>> sum of powers of odd integers, is indeed known. Search
>> for "Swiss-knife function". It's quite interesting.
>
> If you mean "The Canonical Generating Function or CGF(z) -- a Swiss-knife
> function", then it looks a load of codswallop. Then again quoting Chaitin
> right at the start was bound to prejudice me against it. I gave up before
> I got to anything lucid.
>
> Phil

Actually, Phil, I wasn't referring to Huen's work, which I'll
agree is a bit impenetrable. Instead, I was talking about
Peter Luschny's 2008 result on what might more properly be
called Swiss-knife _polynomials_ (though you can still get
this by having Google search for "Swiss-knife functions";
it's just a bit farther down the list).

Sorry for any confusion.


Regards,

Rick
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