jmc

We need to find a pattern in $T$ first. You may have heard of it by the name Fibonacci sequence. Reduced modulo $7$ (we can still use the recurrence relation), it looks like $$T\equiv \{0,1,1,2,3,5,1,6,0,6,6,5,4,2,6,1,0,1\dots \}.$$The first $16$ terms are $\{0,1,1,2,3,5,1,6,0,6,6,5,4,2,6,1\}.$ As the next two are $0$ and $1$ and since the sequence is defined by recursion on the most recent two terms, the Fibonacci sequence modulo $7$ consists of repetitions of $0,$ $1,$ $1,$ $2,$ $3,$ $5,$ $1,$ $6,$ $0,$ $6,$ $6,$ $5,$ $4,$ $2,$ $6,$ $1.$ Now $$\{\begin{array}{l}a\equiv 5(\mathrm{mod}16)⟹t_a\equiv 5(\mathrm{mod}7)\\ b\equiv 10(\mathrm{mod}16)⟹t_b\equiv 6(\mathrm{mod}7)\\ c\equiv 15(\mathrm{mod}16)⟹t_c\equiv 1(\mathrm{mod}7)\end{array}.$$Thus, $$t_a+t_b+t_c\equiv 5+6+1\equiv 12\equiv \boxed{5}(\mathrm{mod}7).$$