Ex - Fibonacci: Revisited

AtCoder

IDabc300_h

Time6000ms

Memory256MB

Difficulty—

We define the general term $a_n$ of a sequence $a_0, a_1, a_2, \dots$ by: $a_n = \begin{cases} 1 & (0 \leq n \lt K) \\ \displaystyle{\sum_{i=1}^K} a_{n-i} & (K \leq n). \\ \end{cases}$ Given an integer $N$, find the sum, modulo $998244353$, of $a_m$ over all non-negative integers $m$ such that $m\text{ AND }N = m$. ($\text{AND}$ denotes the bitwise AND.) What is bitwise AND? The bitwise AND of non-negative integers $A$ and $B$, $A\text{ AND }B$, is defined as follows. ・When $A\text{ AND }B$ is written in binary, its $2^k$s place ($k \geq 0$) is $1$ if $2^k$s places of $A$ and $B$ are both $1$, and $0$ otherwise. ## Constraints * $1 \leq K \leq 5 \times 10^4$ * $0 \leq N \leq 10^{18}$ * $N$ and $K$ are integers. ## Input The input is given from Standard Input in the following format: $K$ $N$ [samples]

Samples

Input #1

2 6

Output #1

21

$a_0$ and succeeding terms are $1, 1, 2, 3, 5, 8, 13, 21, \dots$. Four non-negative integers, $0, 2, 4$, and $6$, satisfy $6 \text{ AND } m = m$, so the answer is $1 + 2 + 5 + 13 = 21$.

Input #2

2 8

Output #2

Input #3

1 123456789

Output #3

Input #4

300 20230429

Output #4

125461938

Input #5

42923 999999999558876113

Output #5

300300300

API Response (JSON)

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  "problem": {
    "name": "Ex - Fibonacci: Revisited",
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      "content": "We define the general term $a_n$ of a sequence $a_0, a_1, a_2, \\dots$ by: $a_n = \\begin{cases} 1 & (0 \\leq n \\lt K) \\\\ \\displaystyle{\\sum_{i=1}^K} a_{n-i} & (K \\leq n). \\\\ \\end{cases}$    Given an ",
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      "statement_type": "Markdown",
      "content": "We define the general term $a_n$ of a sequence $a_0, a_1, a_2, \\dots$ by:\n\n$a_n = \\begin{cases} 1 & (0 \\leq n \\lt K) \\\\ \\displaystyle{\\sum_{i=1}^K} a_{n-i} & (K \\leq n). \\\\ \\end{cases}$\n\n  \n\nGiven an ...",
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