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        riddles >> medium >> Sum Equals Perfect Square
        
(Message started by: ThudanBlunder on Dec 23^rd, 2008, 8:21am)

Title: Sum Equals Perfect Square
Post by ThudanBlunder on Dec 23^rd, 2008, 8:21am

For which values of n is 1³ - 2³ + 3³ - 4³ + 5³ -....... + (-1)ⁿ⁺¹n³ a perfect square?

Title: Re: Sum Equals Perfect Square
Post by towr on Dec 23^rd, 2008, 8:54am

I suppose it's not entirely proper to first generate a bit of the sequence and look it up; but nevertheless: A001844 (http://www.research.att.com/~njas/sequences/A001844).

So, a bit more along the lines of intent:
[hide]We only need to look at odd n, because otherwise it's negative, and thus not a perfect square.
(2n+1)³-(2n)³ = 1 + 6*n + 12*n²
So we have sum_i=0^(n-1)/2(1 + 6*i + 12*i²) = (n+1)/2 + 6*(n-1)/2((n-1)/2+1)/2+ 12 * (n-1)/2((n-1)/2+1)(2(n-1)/2+1)/6 = (n+1)²*(2*n - 1)/4

Which is a perfect square when 2*n - 1 is.
And of course, those n are easy to find starting with the odd squares; just add one and divide by two. [/hide]

Title: Re: Sum Equals Perfect Square
Post by SMQ on Dec 23^rd, 2008, 9:10am

on 12/23/08 at 08:54:56, towr wrote:

I suppose it's not entirely proper to first generate a bit of the sequence and look it up; but nevertheless: A001844 (http://www.research.att.com/~njas/sequences/A001844).

Hmm, strange that there's no cross reference between that and A011934 (http://www.research.att.com/~njas/sequences/A011934).

--SMQ