The Fibonacci sequence $(f_i)$ is the unique sequence such that

• $f_0=0$
• $f_1=1$
• $f_{i+1}=f_i+f_{i-1}$

Similarly, there is a unique function $A(m,n)$ such that

• $A(0,0)=0$
• $A(0,1)=1$
• $A(m+1,n)=A(m,n+1)+A(m,n)$
• $A(m+1,n+1)=2A(m+1,n)+A(m,n)$

Define $S(k)=\displaystyle\sum_{i=1}^k A(f_i,f_i)$. You are given $S(5)=7200$.

Find $S(11^N)$, giving your answer modulo $1123581313$.