Schroeder_Numbers_List — Sequences

Computes the first \e n values of the Schroeder sequence.

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Interface

#include <codecogs/maths/combinatorics/sequences/schroeder_numbers_list.h>

using namespace Maths::Combinatorics::Sequences;

std::vector<int> schroeder_numbers_list(int n)

This function generates an array with the first n values in the Schroeder sequence, given by the following recurrent formula

$$S(n) = \frac{1}{n} ( (6n - 9) S(n-1) - (n-3)S(n-2) )$$

(1)

with initial values

$$S(1) = S(2) = 1$$

(2)

The Schroeder sequence is also defined by

$$S(n) = \frac{P(n)(3) - 3P(n-1)(3)}{4(n-1)}$$

(3)

where $P(n)(X)$ is the Legendre polynomial of order $n$.

Example:

#include <codecogs/maths/combinatorics/sequences/schroeder_numbers_list.h>
#include <iostream>
int main() {
  std::vector<int> result = Maths::Combinatorics::Sequences::schroeder_numbers_list(10);
  std::cout << "Number of values: " << result.size() << std::endl;
  for (int i = 0; i < result.size(); i++)
    std::cout << result[i] << "  ";
  std::cout << std::endl;
  return 0;
}

Output:

Number of values: 10
1  1  3  11  45  197  903  4279  20793  103049

References

SUBSET, a C++ library of combinatorial routines, http://www.csit.fsu.edu/~burkardt/cpp_src/subset/subset.html

Parameters

the number of Schroeder numbers to generate

Returns

an array of the first n values in the Schroeder sequence