21165번 - Redundant Binary Notation 다국어

문제

Binomial trees of a binomial heap. Wikimedia, cc-by-sa

Redundant binary notation is similar to binary notation, except instead of allowing only 0ドル$’s and 1ドル$’s for each digit, we allow any integer digit in the range $[0, t],ドル where $t$ is some specified upper bound. For example, if $t = 2,ドル the digit 2ドル$ is permitted, and we may write the decimal number 4ドル$ as 100ドル,ドル 20ドル,ドル or 12ドル$. If $t=1,ドル every number has precisely one representation, which is its typical binary representation. In general, if a number is written as $d_l d_{l-1} \ldots d_1 d_0$ in redundant binary notation, the equivalent decimal number is $d_l\cdot2^l + d_{l-1}\cdot2^{l-1} + \cdots + d_1\cdot2^1 + d_0\cdot2^0$.

Redundant binary notation can allow carryless arithmetic, and thus has applications in hardware design and even in the design of worst-case data structures. For example, consider insertion into a standard binomial heap. This operation takes $O(\log n)$ worst-case time but $O(1)$ amortized time. This is because the binary number representing the total number of elements in the heap can be incremented in $O(\log n)$ worst-case time and $O(1)$ amortized time. By using a redundant binary representation of the individual binomial trees in a binomial heap, it is possible to improve the worst-case insertion time of binomial heaps to $O(1)$.

However, none of that information is relevant to this question. In this question, your task is simple. Given a decimal number $N$ and the digit upper bound $t,ドル you are to count the number of possible representations $N$ has in redundant binary notation with each digit in range $[0, t]$ with no leading zeros.

입력

Input consists of a single line with two decimal integers $N$ (0ドル \leq N \leq 10^{16}$) and $t$ (1ドル \leq t \leq 100$).

출력

Output in decimal the number of representations the decimal number $N$ has in redundant binary notation with each digit in range $[0, t]$ with no leading zeros. Since the number of representations may be very large, output the answer modulo the large prime 998ドル,244円,353円$.

제한

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