It is possible to find positive integers $A$ and $B$ such that given any triangular number, $T_n$, then $A{T}_n+B$ is always a triangular number. We define $F_3(N)$ to be the sum of $(A+B)$ over all such possible pairs $(A,B)$ with $max(A,B)\le N$. For example $F_3(100)=184$.

Polygonal numbers are generalisations of triangular numbers. Polygonal numbers with parameter $k$ we call $k$-gonal numbers. The formula for the $n$th $k$-gonal number is $\frac{1}{2}n(n(k-2)+4-k)$ where $n\ge 1$. For example when $k=3$ we get $\frac{1}{2}n(n+1)$ the formula for triangular numbers.

The statement above is true for pentagonal, heptagonal and in fact any $k$-gonal number with $k$ odd. For example when $k=5$ we get the pentagonal numbers and we can find positive integers $A$ and $B$ such that given any pentagonal number, $P_n$, then $A{P}_n+B$ is always a pentagonal number. We define $F_5(N)$ to be the sum of $(A+B)$ over all such possible pairs $(A,B)$ with $max(A,B)\le N$.

Similarly we define $F_k(N)$ for odd $k$. You are given $\sum _k{F}_k({10}^3)=14993$ where the sum is over all odd $k=3,5,7,\dots$.

Find $\sum _k{F}_k({10}^{12})$ where the sum is over all odd $k=3,5,7,\dots$