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Topic: Sum Equals Perfect Square (Read 585 times) |
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ThudnBlunder
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Sum Equals Perfect Square
« on: Dec 23rd, 2008, 8:21am » |
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For which values of n is 13 - 23 + 33 - 43 + 53 -....... + (-1)n+1n3 a perfect square?
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towr
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Re: Sum Equals Perfect Square
« Reply #1 on: Dec 23rd, 2008, 8:54am » |
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I suppose it's not entirely proper to first generate a bit of the sequence and look it up; but nevertheless: A001844.
So, a bit more along the lines of intent:
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 sumi=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.
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| « Last Edit: Dec 23rd, 2008, 9:01am by towr » |
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Wikipedia, Google, Mathworld, Integer sequence DB
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SMQ
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Re: Sum Equals Perfect Square
« Reply #2 on: Dec 23rd, 2008, 9:10am » |
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on Dec 23rd, 2008, 8:54am, towr wrote:| I suppose it's not entirely proper to first generate a bit of the sequence and look it up; but nevertheless: A001844. |
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Hmm, strange that there's no cross reference between that and A011934.
--SMQ
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| « Last Edit: Dec 23rd, 2008, 9:12am by SMQ » |
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