Problem 713

Turán’s water heating system

Turán has the electrical water heating system outside his house in a shed. The electrical system uses two fuses in series, one in the house and one in the shed. (Nowadays old fashioned fuses are often replaced with reusable mini circuit breakers, but Turán’s system still uses old fashioned fuses.) For the heating system to work both fuses must work.

Turán has $N$ fuses. He knows that $m$ of them are working and the rest are blown. However, he doesn’t know which ones are blown. So he tries different combinations until the heating system turns on.
We denote by $T (N, m)$ the smallest number of tries required to ensure the heating system turns on.
$T (3, 2) = 3$ and $T (8, 4) = 7$ .

Let $L (N)$ be the sum of all $T (N, m)$ for $2 \leq m \leq N$ .
$L (10^{3}) = 3281346$ .

Find $L (10^{7})$ .

图兰的电热水器

图兰的电热水器装在他家房子旁的木棚里。这套电热水器需要用到两根保险丝，房子里和木棚里各有一根。（如今这些过时的保险丝都被可重复使用的迷你断路器所替代，但是图兰的电热水器仍在用这些保险丝。）只有两根保险丝都工作时，电热水器才能正常使用。

图兰有 $N$ 根保险丝，他知道这其中有 $m$ 根仍能够工作而剩下的都坏了。然而，他并不知道哪些是坏的。因此，他依次尝试所有可能的组合直到打开电热水器为止。
记 $T (N, m)$ 为确保电热水器能够工作所需的最少尝试次数。
已知 $T (3, 2) = 3$ ， $T (8, 4) = 7$ 。

记 $L (N)$ 为所有 $2 \leq m \leq N$ 的 $T (N, m)$ 之和。
已知 $L (10^{3}) = 3281346$ 。

求 $L (10^{7})$ 。