(I start by saying that the tags are probably not accurate but I didn't knwo what to put, so if someone knows what I could tag this question with, let me know in the comments and I'll provide to edit it)

Fix $n$ and consider $n$ variables $x_1 ,x_2 , \dots, x_n$ such that $x_i \geq 1$ for every $i$ and

$$ \sum_{i=1} ^n x_i =n + C$$

for some $C>0$ independent of $n$. Consider now the function:

$$ f(x_1, x_2 , \dots , x_n) = x_{n} x_{n-1} \dots x_1 + x_{n} x_{n-1} \dots x_2 + \dots x_{n}x_{n-1} + x_n .$$

The question is: is there some "simple" expression involving $n$ and $C$ to find $\max (f_n)$ and $argmax(f_n)$ ? Even at the limit for $n \to + \infty$? I suspect that keeping $C$ fixed and increasing $n$ should have a similar effect to keeping $n$ fixed and decreasing $C$, i.e the values of $x_i$ for the argmax should all tend to $1 + C/n$, although I'm not sure how to prove this.

There's also another strange fact, at least not one I can immediately explain. Keeping $n$ fixed and increasing $C$, it appears that the maximum is achieved for $(x_1, x_2, \dots, x_n)=(m, m+1, m+1 , \dots , m+1)$, with $m$ a suitable number so that the constraint is true. No idea why it happens, I would have bet on $x_i = x_j$ for every $i,j$. I tried this on Mathematica for $n=3,6,9$ with $C$ around 10000.

EDIT: I should probably say why I am interested in this, maybe it's something known. If you consider the sequence

$$ \begin{cases} x_{n+1} = \alpha x_n + 1\\ x_0 =0 \end{cases}$$

it's trivial to see that after $N$ steps you end up with something like $\sum_{j=0} ^{N-1} \alpha ^j$, which everyone knows how to treat. But what if the $\alpha= \alpha_n$ depends on $n$ and you only know what's the value of $\sum_{n=0} ^{N-1} \alpha_n $?

(I start by saying that the tags are probably not accurate but I didn't know what to put, so if someone knows what I could tag this question with, let me know in the comments and I'll provide to edit it)

Fix $n$ and consider $n$ variables $x_1 ,x_2 , \dots, x_n$ such that $x_i \geq 1$ for every $i$ and

$$ \sum_{i=1} ^n x_i =n + C$$

for some $C>0$ independent of $n$. Consider now the function:

$$ f(x_1, x_2 , \dots , x_n) = x_{n} x_{n-1} \dots x_1 + x_{n} x_{n-1} \dots x_2 + \dots x_{n}x_{n-1} + x_n .$$

The question is: is there some "simple" expression involving $n$ and $C$ to find $\max (f_n)$ and $argmax(f_n)$ ? Even at the limit for $n \to + \infty$? I suspect that keeping $C$ fixed and increasing $n$ should have a similar effect to keeping $n$ fixed and decreasing $C$, i.e the values of $x_i$ for the argmax should all tend to $1 + C/n$, although I'm not sure how to prove this.

There's also another strange fact, at least not one I can immediately explain. Keeping $n$ fixed and increasing $C$, it appears that the maximum is achieved for $(x_1, x_2, \dots, x_n)=(m, m+1, m+1 , \dots , m+1)$, with $m$ a suitable number so that the constraint is true. No idea why it happens, I would have bet on $x_i = x_j$ for every $i,j$. I tried this on Mathematica for $n=3,6,9$ with $C$ around 10000.

EDIT: I should probably say why I am interested in this, maybe it's something known. If you consider the sequence

$$ \begin{cases} x_{n+1} = \alpha x_n + 1\\ x_0 =0 \end{cases}$$

it's trivial to see that after $N$ steps you end up with something like $\sum_{j=0} ^{N-1} \alpha ^j$, which everyone knows how to treat. But what if the $\alpha= \alpha_n$ depends on $n$ and you only know what's the value of $\sum_{n=0} ^{N-1} \alpha_n $?

(I start by saying that the tags are probably not accurate but I didn't knwo what to put, so if someone knows what I could tag this question with, let me know in the comments and I'll provide to edit it)

Fix $n$ and consider $n$ variables $x_1 ,x_2 , \dots, x_n$ such that $x_i \geq 1$ for every $i$ and

$$ \sum_{i=1} ^n x_i =n + C$$

for some $C>0$ independent of $n$. Consider now the function:

$$ f(x_1, x_2 , \dots , x_n) = x_{n} x_{n-1} \dots x_1 + x_{n} x_{n-1} \dots x_2 + \dots x_{n}x_{n-1} + x_n .$$

The question is: is there some "simple" expression involving $n$ and $C$ to find $\max (f_n)$ and $argmax(f_n)$ ? Even at the limit for $n \to + \infty$? I suspect that keeping $C$ fixed and increasing $n$ should have a similar effect to keeping $n$ fixed and decreasing $C$, i.e the values of $x_i$ for the argmax should all tend to $1 + C/n$, although I'm not sure how to prove this.

There's also another strange fact, at least not one I can immediately explain. Keeping $n$ fixed and increasing $C$, it appears that the maximum is achieved for $(x_1, x_2, \dots, x_n)=(m, m+1, m+1 , \dots , m+1)$, with $m$ a suitable number so that the constraint is true. No idea why it happens, I would have bet on $x_i = x_j$ for every $i,j$. I tried this on Mathematica for $n=3,6,9$ with $C$ around 10000.

(I start by saying that the tags are probably not accurate but I didn't knwo what to put, so if someone knows what I could tag this question with, let me know in the comments and I'll provide to edit it)

Fix $n$ and consider $n$ variables $x_1 ,x_2 , \dots, x_n$ such that $x_i \geq 1$ for every $i$ and

$$ \sum_{i=1} ^n x_i =n + C$$

for some $C>0$ independent of $n$. Consider now the function:

$$ f(x_1, x_2 , \dots , x_n) = x_{n} x_{n-1} \dots x_1 + x_{n} x_{n-1} \dots x_2 + \dots x_{n}x_{n-1} + x_n .$$

The question is: is there some "simple" expression involving $n$ and $C$ to find $\max (f_n)$ and $argmax(f_n)$ ? Even at the limit for $n \to + \infty$? I suspect that keeping $C$ fixed and increasing $n$ should have a similar effect to keeping $n$ fixed and decreasing $C$, i.e the values of $x_i$ for the argmax should all tend to $1 + C/n$, although I'm not sure how to prove this.

There's also another strange fact, at least not one I can immediately explain. Keeping $n$ fixed and increasing $C$, it appears that the maximum is achieved for $(x_1, x_2, \dots, x_n)=(m, m+1, m+1 , \dots , m+1)$, with $m$ a suitable number so that the constraint is true. No idea why it happens, I would have bet on $x_i = x_j$ for every $i,j$. I tried this on Mathematica for $n=3,6,9$ with $C$ around 10000.

EDIT: I should probably say why I am interested in this, maybe it's something known. If you consider the sequence

$$ \begin{cases} x_{n+1} = \alpha x_n + 1\\ x_0 =0 \end{cases}$$

it's trivial to see that after $N$ steps you end up with something like $\sum_{j=0} ^{N-1} \alpha ^j$, which everyone knows how to treat. But what if the $\alpha= \alpha_n$ depends on $n$ and you only know what's the value of $\sum_{n=0} ^{N-1} \alpha_n $?