Shortest Lattice Vector

 Published on Sunday, 15th March 2015, 07:00 am; Solved by 198;
Difficulty rating: 90%

Problem 507

Let $t_n$ be the tribonacci numbers defined as:
$t_0 = t_1 = 0$;
$t_2 = 1$;
$t_n = t_{n-1} + t_{n-2} + t_{n-3}$ for $n \ge 3$
and let $r_n = t_n \text{ mod } 10^7$.

For each pair of Vectors $V_n=(v_1,v_2,v_3)$ and $W_n=(w_1,w_2,w_3)$ with $v_1=r_{12n-11}-r_{12n-10}, v_2=r_{12n-9}+r_{12n-8}, v_3=r_{12n-7} \cdot r_{12n-6}$ and
$w_1=r_{12n-5}-r_{12n-4}, w_2=r_{12n-3}+r_{12n-2}, w_3=r_{12n-1} \cdot r_{12n}$
we define $S(n)$ as the minimal value of the manhattan length of the vector $D=k \cdot V_n+l \cdot W_n$ measured as $|k \cdot v_1+l \cdot w_1|+|k \cdot v_2+l \cdot w_2|+|k \cdot v_3+l \cdot w_3|$ for any integers $k$ and $l$ with $(k,l)\neq (0,0)$.

The first vector pair is (-1, 3, 28), (-11, 125, 40826).
You are given that $S(1)=32$ and $\sum_{n=1}^{10} S(n)=130762273722$.

Find $\sum_{n=1}^{20000000} S(n)$.