Lights Out

 Published on Sunday, 22nd March 2020, 10:00 am; Solved by 182;
Difficulty rating: 55%

Problem 707

Consider a $w\times h$ grid. A cell is either ON or OFF. When a cell is selected, that cell and all cells connected to that cell by an edge are toggled on-off, off-on. See the diagram for the 3 cases of selecting a corner cell, an edge cell or central cell in a grid that has all cells on (white).


The goal is to get every cell to be off simultaneously. This is not possible for all starting states. A state is solvable if, by a process of selecting cells, the goal can be achieved.

Let $F(w,h)$ be the number of solvable states for a $w\times h$ grid. You are given $F(1,2)=2$, $F(3,3) = 512$, $F(4,4) = 4096$ and $F(7,11) \equiv 270016253 \pmod{1\,000\,000\,007}$.

Let $f_1=f_2 = 1$ and $f_n=f_{n-1}+f_{n-2}, n \ge 3$ be the Fibonacci sequence and define $$ S(w,n) = \sum_{k=1}^n F(w,f_k)$$ You are given $S(3,3) = 32$, $S(4,5) = 1052960$ and $S(5,7) \equiv 346547294 \pmod{1\,000\,000\,007}$.

Find $S(199,199)$. Give your answer modulo $1\,000\,000\,007$.