Question 108
Find the number of binary sequences $S$ of length $2015$ such that for any two segments $I_{1}, I_{2}$ of $S$ of the same length, we have
- The sum of digits of $I_{1}$ differs from the sum of digits of $I_{2}$ by at most $1$;
- If $I_{1}$ begins on the left end of $S$ then the sum of digits of $I_{1}$ is not greater than the sum of digits of $I_{2}$;
- If $I_{2}$ ends on the right end of $S$ then the sum of digits of $I_{2}$ is not less than the sum of digits of $I_{1}$.
- The sum of digits of $I_{1}$ differs from the sum of digits of $I_{2}$ by at most $1$;
- If $I_{1}$ begins on the left end of $S$ then the sum of digits of $I_{1}$ is not greater than the sum of digits of $I_{2}$;
- If $I_{2}$ ends on the right end of $S$ then the sum of digits of $I_{2}$ is not less than the sum of digits of $I_{1}$.