Math Is Fun Forum

Nehushtan

Member

Registered: 2013-03-09

Posts: 957

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!

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 (2013-11-23 17:01:09)

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