Jack's Bean (noch nicht übersetzt)

Jack has three plates in front of him. The giant has $N$ beans that he distributes to the three plates. All the beans look the same, but one of them is a magic bean. Jack doesn't know which one it is, but the giant knows.

Jack can ask the giant questions of the form: "Does this subset of the beans contain the magic bean?" In each question Jack may choose any subset of beans from a single plate, and the giant will respond truthfully.

If the three plates contain $a$, $b$ and $c$ beans respectively, we let $h(a, b, c)$ be the minimal number of questions Jack needs to ask in order to guarantee he locates the magic bean. For example, $h(1, 2, 3) = 3$ and $h(2, 3, 3) = 4$.

Let $H(N)$ be the sum of $h(a, b, c)$ over all triples of non-negative integers $a$, $b$, $c$ with $1 \leq a + b + c \leq N$.
You are given: $H(6) = 203$ and $H(20) = 7718$.

A repunit, $R_n$, is a number made up with $n$ digits all '1'. For example, $R_3 = 111$ and $H(R_3) = 1634144$.

Find $H(R_{19})$. Give your answer modulo $1\,000\,000\,007$.