Palindrome Basis

2000ms

262144K

Description:

You are given a positive integer $$$n$$$. Let's call some positive integer $$$a$$$ without leading zeroes palindromic if it remains the same after reversing the order of its digits. Find the number of distinct ways to express $$$n$$$ as a sum of positive palindromic integers. Two ways are considered different if the frequency of at least one palindromic integer is different in them. For example, $$$5=4+1$$$ and $$$5=3+1+1$$$ are considered different but $$$5=3+1+1$$$ and $$$5=1+3+1$$$ are considered the same.

Formally, you need to find the number of distinct multisets of positive palindromic integers the sum of which is equal to $$$n$$$.

Since the answer can be quite large, print it modulo $$$10^9+7$$$.

Input:

The first line of input contains a single integer $$$t$$$ ($$$1\leq t\leq 10^4$$$) denoting the number of testcases.

Each testcase contains a single line of input containing a single integer $$$n$$$ ($$$1\leq n\leq 4\cdot 10^4$$$) — the required sum of palindromic integers.

Output:

For each testcase, print a single integer denoting the required answer modulo $$$10^9+7$$$.

Sample Input:

2
5
12

Sample Output:

7
74

Note:

For the first testcase, there are $$$7$$$ ways to partition $$$5$$$ as a sum of positive palindromic integers:

$$$5=1+1+1+1+1$$$
$$$5=1+1+1+2$$$
$$$5=1+2+2$$$
$$$5=1+1+3$$$
$$$5=2+3$$$
$$$5=1+4$$$
$$$5=5$$$

For the second testcase, there are total $$$77$$$ ways to partition $$$12$$$ as a sum of positive integers but among them, the partitions $$$12=2+10$$$, $$$12=1+1+10$$$ and $$$12=12$$$ are not valid partitions of $$$12$$$ as a sum of positive palindromic integers because $$$10$$$ and $$$12$$$ are not palindromic. So, there are $$$74$$$ ways to partition $$$12$$$ as a sum of positive palindromic integers.

Informação

Provedor Codeforces

Código CF1673C