# [seqfan] x(i-1) = ( x(i+1) - x(i)) mod M, period-N solutions?

Ron Hardin rhhardin at att.net
Sat Sep 4 14:41:13 CEST 2010

```T(N,M) is the number of nonzero solutions of  x(i-1) = (x(i+1) - x(i)) mod M,
with period N

for N=12 M=10 there are 19 (including shifted multiplicities)
1   0 5 5 0 5 5 0 5 5 0 5 5
2   5 0 5 5 0 5 5 0 5 5 0 5
3   5 5 0 5 5 0 5 5 0 5 5 0
4   1 3 4 7 1 8 9 7 6 3 9 2
5   1 8 9 7 6 3 9 2 1 3 4 7
6   6 3 9 2 1 3 4 7 1 8 9 7
7   6 8 4 2 6 8 4 2 6 8 4 2
8   2 1 3 4 7 1 8 9 7 6 3 9
9   2 6 8 4 2 6 8 4 2 6 8 4
10   7 1 8 9 7 6 3 9 2 1 3 4
11   7 6 3 9 2 1 3 4 7 1 8 9
12   4 2 6 8 4 2 6 8 4 2 6 8
13   4 7 1 8 9 7 6 3 9 2 1 3
14   9 2 1 3 4 7 1 8 9 7 6 3
15   9 7 6 3 9 2 1 3 4 7 1 8
16   3 4 7 1 8 9 7 6 3 9 2 1
17   3 9 2 1 3 4 7 1 8 9 7 6
18   8 4 2 6 8 4 2 6 8 4 2 6
19   8 9 7 6 3 9 2 1 3 4 7 1

T(N,M) starts  (rows N=1..  cols M=1..)
0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
0   3   0   3   0   3   0   3   0   3   0   3   0   3   0   3   0   3   0
0   0   0   0   4   0   0   0   0   4   0   0   0   0   4   0   0   0   0
0   0   0   0   0   0   0   0   0   0  10   0   0   0   0   0   0   0   0
0   3   0  15   0   3   0  15   0   3   0  15   0   3   0  15   0   3   0
0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
0   0   8   0   4   8   0   0   8   4   0   8   0   0  44   0   0   8   0
0   3   0   3   0   3   0   3   0   3   0   3   0   3   0   3   0   3  18
0   0   0   0   0   0   0   0   0   0 120   0   0   0   0   0   0   0   0
0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
0   3   0  15   4   3   0  63   0  19   0  15   0   3   4  63   0   3   0
0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
0   3   0   3   0   3   0   3   0   3  10   3   0   3   0   3   0   3   0
0   0   8   0   4   8  48   0   8   4   0   8   0  48  44   0   0   8   0
0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
0   3   0  15   0   3   0  15   0   3   0  15   0   3   0  15   0   3 360
0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
0   0   0   0  24   0   0   0   0  24 120   0   0   0  24   0   0   0   0

full current table occasionally updated at
http://rhhardin.home.mindspring.com/current5.txt

Ideas, formulas, is it already in OEIS in some form...?

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

```