I have been playing around with interesting integer sequences and came across [Schröder number][1] which defines the number of lattice paths of n x n grid.

The recurrence formula to calculate these numbers is as follows:

$$
S_n = 3S_{n-1} + \sum_{k=1}^{n-2}S_kS_{n-k-1} \text{  for  }n\ge2 \text{ with } S_0 =1, S_2=2 
$$
The first few numbers in the sequence are:
$$
1, 2, 6, 22, 90, 394, 1806, 8558
$$
I was curious about the following ratio and its limit:
$$
\lim_{n\rightarrow\infty}\frac{S_{n-1}}{S_{n}} = ? 
$$
So I wrote a small C++ program with mixed-precision to handle the large numbers and surprisingly there appears to be an asymptote that is not zero. So far my little program has reached n = 20,000, but it's slowing down significantly as the numbers are getting truly large but it has given a preliminary result of
$$
\lim_{n\rightarrow\infty}\frac{S_{n-1}}{S_{n}} = 0.17158... 
$$

My question now is, is this simply a ghost limit for not having gone far enough, or does a limit exist that is not zero? My math expertise is lacking somewhat to attempt to find that limit analytically (can it even be done?).

If a limit exists would it imply that, for large n, the number of additional lattice paths of a n x n grid only increases by about 17% by expanding the grid by 1?

Code for anyone who wants to try it for themselves:
```code
#include <iostream>
#include <gmp.h>
#include <gmpxx.h>
#include <vector>

void schroder(std::vector<mpz_class>& n_vec);

int main(int argc, char **argv)
{	
	int n=2;
	int n_max;
	std::cout << "Enter n_max: ";
	std::cin >> n_max;
	std::vector<mpz_class> s = {mpz_class(1), mpz_class(2)};
	while (n<n_max){
		mpq_class div (s[s.size()-2],s.back());
		gmp_printf ("%d\t %.*Ff\n", n, 100, mpf_class(div,500));
		schroder(s);
		n++;
	}
	return 0;
}

void schroder(std::vector<mpz_class>& n_vec){
	int n = n_vec.size();
	mpz_class sum = mpz_class(0);
	for(int k = 1; k<=(n-2); k++){
		sum += n_vec[k]*n_vec[n-k-1];
	}
	sum += 3*n_vec[n-1];
	n_vec.push_back(sum);
}
```



  [1]: https://en.wikipedia.org/wiki/Schr%C3%B6der_number