# Problem 507: Shortest Lattice Vector
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)\$.