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#1 2013-11-24 15:46:20

Nehushtan
Power Member

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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 it’s just 12345678910987654321 with a carry-over from the 10 in the middle. This leads to the following conjecture:



Let’s prove it! big_smile

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 (2013-11-24 16:01:09)


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