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You are not logged in. #1 20131124 15:46:20
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 it’s just 12345678910987654321 with a carryover from the 10 in the middle. This leads to the following conjecture: Let’s 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. Hence . Last edited by Nehushtan (20131124 16:01:09) 134 books currently added on Goodreads 