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

[Ron Hardin]

Thanks, the following (awk) formula for tab(i,j) seems to work for rows 1..79 so far

if(int((i+1)/2)%2!=int(j/2)%2)ans=1;
else if((i-j)%2==0) {
        n=(i-j)/2;
        if(j>n^2)ans=1;
        else ans=0;
}
else ans=0;

[int() is floor, % is modulo]




 rhhardin at mindspring.com
rhhardin at att.net (either)



----- Original Message ----
> From: "franktaw at netscape.net" <franktaw at netscape.net>
> To: seqfan at list.seqfan.eu
> Sent: Sun, March 21, 2010 11:03:25 PM
> Subject: [seqfan] Re: Sum +-j +-(j+1) +-(j+2) ... +-i equals zero.
> 
> I think you would get a better idea of what is happening if you were to 
show 
> a table of the minimum possible non-negative value of the sum, 
instead of 
> just zeros and ones.

The blocks of 2x2 zeros and ones reflect parity: the 
> sum has the same 
parity regardless of what signs are used, so when that 
> parity is odd, 
there can be no solution.

The stuff happening near the 
> diagonal reflects the cases where it the 
individual numbers are too big for 
> the variation between numbers to 
offset them. Cases where j-i is even have 
> an odd number of terms, and 
you will see increasing minimums on such 
> diagonals. Where i-j+1 == 2 
(mod 4), the parity will always be wrong.  
> When i-j+1 == 0 (mod 4), 
there is always a solution: +j-(j+1)-(j+2)+(j+3) 
> 
+(j+4)-(j+5)-(j+6)+(j+7) + ....

In order to get a solution with an 
> odd number of terms 2n+1 (so i = j + 
2n), you need the first n+1 terms to 
> sum to at least the sum of the 
last n terms: j(n+1)+n(n+1)/2 >= jn 
> +n(3n+1)/2, which reduces to j >= 
n^2.

I think these two 
> considerations cover every case, although I haven't 
spent the time to fully 
> work out the proof.

Franklin T. Adams-Watters

-----Original 
> Message-----
From: Ron Hardin <
> href="mailto:rhhardin at att.net">rhhardin at att.net>

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
08  0 1 1 0 0 1 1 1
09  1 0 0 1 1 0 1 1 1
10  1 0 0 1 
> 1 1 0 1 1 1
11  0 1 1 0 0 1 1 0 1 1 1
12  0 1 1 0 0 1 1 1 0 1 1 
> 1
13  1 0 0 1 1 0 0 1 1 0 1 1 1
14  1 0 0 1 1 0 0 1 1 1 0 1 1 
> 1
15  0 1 1 0 0 1 1 0 0 1 1 0 1 1 1
16  0 1 1 0 0 1 1 0 0 1 1 1 
> 0 1 1 1
17  1 0 0 1 1 0 0 1 1 0 1 1 1 0 1 1 1
18  1 0 0 1 1 0 0 
> 1 1 0 0 1 1 1 0 1 1 1
19  0 1 1 0 0 1 1 0 0 1 1 0 1 1 1 0 1 1 
> 1
20  0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 1 1 1
21  1 0 0 1 1 0 0 
> 1 1 0 0 1 1 0 1 1 1 0 1 1 1
22  1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 1 
> 1 1
23  0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 1 1 0 1 1 1
24  0 1 1 
> 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 1 1 1
25  1 0 0 1 1 0 0 1 1 0 0 1 1 
> 0 0 1 1 0 1 1 1 0 1 1 1
26  1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 1 1 1 
> 0 1 1 1
27  0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 1 1 0 1 1 
> 1
28  0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 1 1 1 0 1 1 
> 1
29  1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 1 1 0 1 1 
> 1
30  1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 1 1 1 0 1 1 
> 1
31  0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 1 1 0 1 1 
> 1
32  0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 1 1 1 0 1 1 
> 1
33  1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 1 1 0 1 1 
> 1
34  1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 1 1 1 0 1 1 
> 1
35  0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 1 1 0 1 
> 1 
1
36  0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 1 
> 1 1 0 1 
1 1
37  1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 
> 1 1 1 0 1 1 1 0 
1 1 1
38  1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 
> 0 0 1 1 0 0 1 1 1 0 1 1 1 
0 1 1 1
39  0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 
> 0 0 1 1 0 0 1 1 0 0 1 1 0 1 1 1 0 1 1 
1 0 1 1
1
40  0 1 1 0 0 1 1 
> 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 1 
1 1 0 1
1 
> 1

A formula would be nice (that was the original plan - I don't have to 
> 
check for
zero if the sum can't be zero.

The weirdness is the slow 
> replacement of 2x2 zero blocks by a single 
diagonal
zero, 
> encroaching
leftwards from the main 
> diagonal.


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