970. Kangaroo Hopping over Sixes

Starting at zero, a kangaroo hops along the real number line in the positive direction. Each successive hop takes the kangaroo forward a uniformly random distance between $0$$0$ and $1$$1$. Let $H(n)$$H(n)$ be the expected number of hops needed for the kangaroo to pass $n$$n$ on the real line.

For example, $H(2)\approx 4.67077427047$$H(2) \approx 4.67077427047$. The first eight digits after the decimal point that are different from six are $70774270$$70774270$.

Similarly, $H(3)\approx 6.6665656395558899$$H(3) \approx 6.6665656395558899$. Here the first eight digits after the decimal point that are different from six are $55395558$$55395558$.

Find $H({10}^6)$$H(10^6)$ and give as your answer the first eight digits after the decimal point that are different from six.

970. 袋鼠跳过数字六

一只袋鼠正以一数轴的原点为起点，沿着数轴的正半轴方向跳跃。这只袋鼠每次跳跃的距离服从 $[0,1]$$[0, 1]$ 上的均匀分布。记 $H(n)$$H(n)$ 为：袋鼠经过数轴上代表 $n$$n$ 的点时，跳跃次数的期望。

例如，$H(2)\approx 4.67077427047$$H(2) \approx 4.67077427047$，这个数小数点后不是 $6$$6$ 的前八个数位是 $70774270$$70774270$。类似的，有 $H(3)\approx 6.6665656395558899$$H(3) \approx 6.6665656395558899$，其小数点后不是 $6$$6$ 的前八个数位是 $55395558$$55395558$。

求 $H({10}^6)$$H(10^6)$，并给出这个数小数点后不是 $6$$6$ 的前八个数位作为答案。

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