[seqfan] Sum +-j +-(j+1) +-(j+2) ... +-i equals zero.
Ron Hardin
rhhardin at att.net
Mon Mar 22 03:32:04 CET 2010
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.
rhhardin at mindspring.com
rhhardin at att.net (either)
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