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1², 11², 111², 1111²,
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 formula
Given , 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.
Last edited by Nehushtan (2013-11-24 16:01:09)
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