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This is something I discovered on another forum.

When you get to

the pattern appears to be lost but in fact its just 12345678910987654321 with a carry-over from the 10 in the middle. This leads to the following conjecture:Lets prove it!

Now

using the formulaGiven

, how many pairs are there with such that ?Suppose

is even. Then so there are such pairs. Thus the coefficent of is .Suppose

is odd. Then so again there are such pairs. Then the coefficent of is again.For

, we let ; then giving pairs. Since we have ; therefore or . If is even, then so is ; then and the coefficient of is . If is odd, then so is ; so and the coefficient of is again.Hence

.*Last edited by Nehushtan (2013-11-23 17:01:09)*

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