This is something I discovered on another forum.
When you get tothe 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!
Nowusing the formula
Given, how many pairs are there with such that ?
Supposeis even. Then so there are such pairs. Thus the coefficent of is .
Supposeis 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 (2013-11-23 17:01:09)