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August 29th, 2013, 03:57 PM   #11
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Re: Express a cube as sum of squares ....

Quote:
 Originally Posted by CRGreathouse If you're allowed unbounded sums there are thousands of identities. You can check this one with Faulhaber's formula, if you like.
There is no link to Faulhaber's formula.

It is about combinatorials not sum of powers

August 29th, 2013, 04:00 PM   #12
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Re: Express a cube as sum of squares ....

Quote:
Originally Posted by icemanfan
Quote:
 Originally Posted by Mouhaha Sorry. Im talking about a general formula. n^3 is expressed by ns (squares and degree 1)
There is no formula in terms of a quadratic function that would work for all n. This is what I meant when I said the only way to represent $n^3$ in terms of n is $n^3$.
Look at my way to write n^3

August 29th, 2013, 04:13 PM   #13
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Re: Express a cube as sum of squares ....

Quote:
Originally Posted by Mouhaha
Quote:
 Originally Posted by CRGreathouse If you're allowed unbounded sums there are thousands of identities. You can check this one with Faulhaber's formula, if you like.
There is no link to Faulhaber's formula.

It is about combinatorials not sum of powers
You're summing a quadratic. If you can't see the link between that and Faulhaber's formula, then it wouldn't help if I explained it.

 August 29th, 2013, 04:14 PM #14 Member   Joined: Apr 2013 Posts: 70 Thanks: 0 Re: Express a cube as sum of squares .... When n is even =2k n^3 = 6*(sigma(2i^2) with i=1 to k-1) +3n^2-2n When is n is odd =2k+1 n^3=6*(sigma((2i+1)^2) with i=0 to k-1) +3n^2-2n In Latex it will be great. (Edited and corrected)
 August 29th, 2013, 04:19 PM #15 Member   Joined: Apr 2013 Posts: 70 Thanks: 0 Re: Express a cube as sum of squares .... Can you rewrite my formulas in Latex? It would help a lot I think.
 August 29th, 2013, 04:23 PM #16 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Express a cube as sum of squares .... Maybe you should grab gp (link in my .sig) and use sumformal() to prove them. (You could also use the WZ method, as found in the free (!) book A = B, but that would take much longer to learn!)
 August 29th, 2013, 04:28 PM #17 Member   Joined: Apr 2013 Posts: 70 Thanks: 0 Re: Express a cube as sum of squares .... 1^2+2^2+3^2+4^2+.....n^2 which is the purpose Faulhaber`s formula is different from what I proposed 1^2+2^2+3^2+4^2+.....n^2 can be expressed as sum of 2 combinatorials C(a,3)+C(b,3)
 August 29th, 2013, 04:30 PM #18 Member   Joined: Apr 2013 Posts: 70 Thanks: 0 Re: Express a cube as sum of squares .... Did you get that there is something new in my formulas? If you did not then ..............
 August 29th, 2013, 04:47 PM #19 Member   Joined: Apr 2013 Posts: 70 Thanks: 0 Re: Express a cube as sum of squares .... 1^2+2^2+3^2+4^2+.....n^2 can be expressed as sum of 2 factorials C(a,3)+C(b,3) For example : 1^2+2^2+3^2+4^2+.....11^2= C(12,3)+C(13,3) General formula for n 1^2+2^2+3^2+4^2+.....n^2=C(n+1,3)+C(n+2,3)
August 29th, 2013, 05:35 PM   #20
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Re: Express a cube as sum of squares ....

Quote:
 Originally Posted by Mouhaha Can you rewrite my formulas in Latex? It would help a lot I think.
For n even:

$n^3= 6\sum_{i=1}^{n-1} 2i^2 + 3n^2 - 2n$

For n odd:

$n^3= 6\sum_{i=0}^{n-1}(2i+1)^2 + 3n^2 - 2n$

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