Two coins of each kind (3)

Given a number $x$ and $n$ different coin values $c_1 \ldots c_n$ , compute in how many ways it is possible to achieve change $x$ by using each value at most twice. Here, two coins with the same value are considered equal.

For example, if $x = 4$ and the available values are 1 and 2, then there are two ways to achieve it: $1 + 1 + 2$ and $2 + 2$ . As another example, if $x = 5$ and the available values are 1, 2, 3, 4 and 5, then there are five ways: $1 + 1 + 3$ , $1 + 2 + 2$ , $1 + 4$ , $2 + 3$ and 5.

Input

Input consists of several cases, with only integer numbers. Every case begins with $x$ and $n$ , followed by $c_1 \ldots c_n$ . Assume $1 \le n \le 15$ , $1 \le c_i \le x \le 10^6$ , and that all $c_i$ are different.

Output

For every case print the number of different ways to achieve change exactly $x$ by using each value at most twice.

Hint

A simply pruned backtracking should be enough.

Problem information

Author: Unknown
Translator: Albert Atserias

Generation: 2026-01-25T22:55:02.912Z