The Frobenius equation is the Diophantine equation $$ a_1 x_1+\dots+a_n x_n=b,$$ where the $a_j$ are positive integers, $b$ is an integer, and a solution $$(x_1, \dots, x_n)$$ must consist of non-negative integers, i.e. $$ x_j \in \mathbb{N} $$ as Natural numbers. For negative $b$, there are no solutions.

• My question: Is there any known formula that counts the number of solutions, by giving $a_1, \dots, a_n$, $b$, and $n$? Let us call this function as $F (a_1, \dots, a_n; b, n)$, what is known for this:

$$F (a_1, \dots, a_n; b, n)=?$$

For all the $a_j=1$, we can simplify the above Frobenius equation to: $$ x_1+\dots+x_n=b, \tag{1}$$ where $b \in \mathbb{Z}^+$ is a positive integer.

• Here is another simpler question: Is there a general formula for Eq.(1) counting all the possible solutions $$(x_1, \dots, x_n)$$
  for given the positive integer $n \in \mathbb{Z}^+$ and $b \in \mathbb{Z}^+$? This should be related to the Partition, but I am not sure the exact forms are known? Say, can we find the total number of soultions as a function $f(n,b)$, and what is $$ f(n,b)=? $$

It seems the answer is known:

$$ f(n,b)= \binom{b+n-1}{n-1}? $$

p.s. Sorry if this question is too simple for number theorists. But please provide me answer and Refs if you already know the answer. Many thanks!

The Frobenius equation is the Diophantine equation $$ a_1 x_1+\dots+a_n x_n=b,$$ where the $a_j$ are positive integers, $b$ is an integer, and a solution $$(x_1, \dots, x_n)$$ must consist of non-negative integers, i.e. $$ x_j \in \mathbb{N} $$ as Natural numbers. For negative $b$, there are no solutions.

• My question: Is there any known formula that counts the number of solutions, by giving $a_1, \dots, a_n$, $b$, and $n$? Let us call this function as $F (a_1, \dots, a_n; b, n)$, what is known for this:

$$F (a_1, \dots, a_n; b, n)=?$$

For all the $a_j=1$, we can simplify the above Frobenius equation to: $$ x_1+\dots+x_n=b, \tag{1}$$ where $b \in \mathbb{Z}^+$ is a positive integer.

• Here is another simpler question: Is there a general formula for Eq.(1) counting all the possible solutions $$(x_1, \dots, x_n)$$
  for given the positive integer $n \in \mathbb{Z}^+$ and $b \in \mathbb{Z}^+$? This should be related to the Partition, but I am not sure the exact forms are known? Say, can we find the total number of soultions as a function $f(n,b)$, and what is $$ f(n,b)=? $$

It seems the answer is known:

$$ f(n,b)= \binom{b+n-1}{n-1}. $$

p.s. Sorry if this question is too simple for number theorists. But please provide me answer and Refs if you already know the answer. Many thanks!

The Frobenius equation is the Diophantine equation $$ a_1 x_1+\dots+a_n x_n=b,$$ where the $a_j$ are positive integers, $b$ is an integer, and a solution $$(x_1, \dots, x_n)$$ must consist of non-negative integers, i.e. $$ x_j \in \mathbb{N} $$ as Natural numbers. For negative $b$, there are no solutions.

For all the $a_j=1$, we can simplify the above Frobenius equation to: $$ x_1+\dots+x_n=b, \tag{1}$$ where $b \in \mathbb{Z}^+$ is a positive integer.

Here is my question: Is there a general formula for Eq.(1) counting all the possible solutions $$(x_1, \dots, x_n)$$
for given the positive integer $n \in \mathbb{Z}^+$ and $b \in \mathbb{Z}^+$? This should be related to the Partition, but I am not sure the exact forms are known? Say, can we find the total number of soultions as a function $f(n,b)$, and what is $$ f(n,b)=? $$

p.s. Sorry if this question is too simple for number theorists. But please provide me answer and Refs if you already know the answer. Many thanks!

The Frobenius equation is the Diophantine equation $$ a_1 x_1+\dots+a_n x_n=b,$$ where the $a_j$ are positive integers, $b$ is an integer, and a solution $$(x_1, \dots, x_n)$$ must consist of non-negative integers, i.e. $$ x_j \in \mathbb{N} $$ as Natural numbers. For negative $b$, there are no solutions.

• My question: Is there any known formula that counts the number of solutions, by giving $a_1, \dots, a_n$, $b$, and $n$? Let us call this function as $F (a_1, \dots, a_n; b, n)$, what is known for this:

$$F (a_1, \dots, a_n; b, n)=?$$

For all the $a_j=1$, we can simplify the above Frobenius equation to: $$ x_1+\dots+x_n=b, \tag{1}$$ where $b \in \mathbb{Z}^+$ is a positive integer.

• Here is another simpler question: Is there a general formula for Eq.(1) counting all the possible solutions $$(x_1, \dots, x_n)$$
  for given the positive integer $n \in \mathbb{Z}^+$ and $b \in \mathbb{Z}^+$? This should be related to the Partition, but I am not sure the exact forms are known? Say, can we find the total number of soultions as a function $f(n,b)$, and what is $$ f(n,b)=? $$

It seems the answer is known:

$$ f(n,b)= \binom{b+n-1}{n-1}? $$

p.s. Sorry if this question is too simple for number theorists. But please provide me answer and Refs if you already know the answer. Many thanks!