[seqfan] Re: Sum +-j +-(j+1) +-(j+2) ... +-i equals zero.

[Olivier Gerard]

Dear Ron,

Your investigation is related to

A063865 <http://www.research.att.com/~njas/sequences/A063865>


Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 0.
{
 0, 0, 2, 2, 0, 0, 8, 14, 0, 0, 70 ...

About which a paper was published in the JIS giving a GF for it.

And also to number of directions in lattices.

I think for my part it is clearer if you present it slightly differently
than you did,
by counting the number of solutions and putting the second variable not as
a fixed point but as an offset

0 0 2 2 0 0 8 14 0 0 70
0 0 0 2 2 0 0 12 20 0 0
0 0 0 2 0 0 6 10 0 0 54
0 0 0 2 2 0 0 10 16 0 0
0 0 0 2 0 0 4 8 0 0 46
0 0 0 2 0 0 0 8 12 0 0
0 0 0 2 0 0 2 8 0 0 38
0 0 0 2 0 0 0 8 10 0 0
0 0 0 2 0 0 2 8 0 0 32
0 0 0 2 0 0 0 8 6 0 0
0 0 0 2 0 0 0 8 0 0 24
0 0 0 2 0 0 0 8 4 0 0
0 0 0 2 0 0 0 8 0 0 20
0 0 0 2 0 0 0 8 2 0 0
0 0 0 2 0 0 0 8 0 0 14
0 0 0 2 0 0 0 8 2 0 0
0 0 0 2 0 0 0 8 0 0 10
0 0 0 2 0 0 0 8 0 0 0
0 0 0 2 0 0 0 8 0 0 6
0 0 0 2 0 0 0 8 0 0 0
0 0 0 2 0 0 0 8 0 0 4
0 0 0 2 0 0 0 8 0 0 0
0 0 0 2 0 0 0 8 0 0 2
0 0 0 2 0 0 0 8 0 0 0
0 0 0 2 0 0 0 8 0 0 2
0 0 0 2 0 0 0 8 0 0 0


You can recover your original table by transposition and replacing positive
entries by 0 and zero entries by 1.

You see that all entries in the lines after the first are degenerescences of
the first row.
Mostly linear in all cases I have examined.

Of course all values are even because inverting all the signs of the sum
give the
same absolute values.



On Mon, Mar 22, 2010 at 05:32, Ron Hardin <rhhardin at att.net> wrote:

> When can the sum of plus or minus consecutive integers be zero?
>
> I get a weird table, entry i,,j zero iff the sum of +-j +-(j+1) +-(j+2) ...
> +-i equals zero for some choice of signs.
>
> 01  1
> 02  1 1
> 03  0 1 1
> 04  0 1 1 1
> 05  1 0 1 1 1
> 06  1 0 0 1 1 1
> 07  0 1 1 0 1 1 1
>
>