## Correlations of the first 1124 sequence

Here is some data on the partial sums $PS_k(m):=\sum_{i=1}^mx_ix_{i+k}$ (with the convention $x_{i+k}=0$ when $i+k>m$) for small values of $k$.

Here is a plot for $k=1,...12$ (click to enlarge)

The final values at $m=1124$ have some noticeable structure:

firstly, all values of $k$ which are a multiple of $3$ are positive while the rest are negative (at least for $k\leq 39$).

then there are some exact results (accidental ones maybe?), denoting $PS_k(1124)$ by its index $k$ we have: $5=11=35$, $29=31$, $15=39$, $6=-8$, $20=-12$.

k;$PS_k(1124)$

1;-259

2;-268

3;239

4;-256

5;-107

6;178

7;-95

8;-178

9;363

10;-160

11;-107

12;210

13;-111

14;-100

15;219

16;-164

17;-181

18;438

19;-193

20;-210

21;303

22;-126

23;-149

24;254

25;-155

26;-184

27;453

28;-164

29;-139

30;236

31;-139

32;-122

33;217

34;-166

35;-107

36;418

37;-195

38;-120

39;219