889. Rational Blancmange

Recall the blancmange function from Problem 226: $T(x)=\sum _{n=0}^{\mathrm{\infty }}{\displaystyle \frac{s(2^{n}x)}{2^n}}$$T(x) = \sum\limits_{n = 0}^\infty\dfrac{s(2^nx)}{2^n}$, where $s(x)$$s(x)$ is the distance from $x$$x$ to the nearest integer.

For positive integers $k,t,r$$k, t, r$, we write

It can be shown that $F(k,t,r)$$F(k, t, r)$ is always an integer.
For example, $F(3,1,1)=42$$F(3, 1, 1) = 42$, $F(13,3,3)=23093880$$F(13, 3, 3) = 23093880$ and $F(103,13,6)\equiv 878922518(\mathrm{mod}1000062031)$$F(103, 13, 6) \equiv 878922518\pmod {1\,000\,062\,031}$.

Find $F({10}^{18}+31,{10}^{14}+31,62)$$F(10^{18} + 31, 10^{14} + 31, 62)$. Give your answer modulo $1000062031$$1\,000\,062\,031$.

889. 有理数上的奶冻函数

考虑 第 226 题 中提及的奶冻函数：$T(x)=\sum _{n=0}^{\mathrm{\infty }}{\displaystyle \frac{s(2^{n}x)}{2^n}}$$T(x) = \sum\limits_{n = 0}^\infty\dfrac{s(2^nx)}{2^n}$，其中 $s(x)$$s(x)$ 指的是 $x$$x$ 到离它最近的整数的距离。

对正整数 $k,t,r$$k, t, r$，记

可以证明 $F(k,t,r)$$F(k, t, r)$ 始终是整数。例如，$F(3,1,1)=42$$F(3, 1, 1) = 42$、$F(13,3,3)=23093880$$F(13, 3, 3) = 23093880$、$F(103,13,6)\equiv 878922518(\mathrm{mod}1000062031)$$F(103, 13, 6) \equiv 878922518\pmod {1\,000\,062\,031}$。

求 $F({10}^{18}+31,{10}^{14}+31,62)$$F(10^{18} + 31, 10^{14} + 31, 62)$ 模 $1000062031$$1\,000\,062\,031$ 的值。

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