{"app":{"locale":"en-GB"},"problem":{"content":"

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

\nFor 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
$w_1=r_{12n-5}-r_{12n-4}, w_2=r_{12n-3}+r_{12n-2}, w_3=r_{12n-1} \\cdot r_{12n}$\n
\n\nwe 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 $|k \\cdot v_1+l \\cdot w_1|+|k \\cdot v_2+l \\cdot w_2|+|k \\cdot v_3+l \\cdot w_3|$\n for any integers $k$ and $l$ with $(k,l)\\neq (0,0)$.\n

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

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

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