Algorithm to efficiently determine the [n][n] element in a matrix

Question

This is a question regarding a piece of coursework so would rather you didn\'t fully answer the question but rather give tips to improve the run time complexity of my curren

Answer 1

Thanks to will's (first) answer, I had this idea:

Consider that any positive solution comes only from the 1's along the x and y axes. Each of the recursive calls to f divides each component of the solution by 3, which means we can sum, combinatorially, how many ways each 1 features as a component of the solution, and consider it's "distance" (measured as how many calls of f it is from the target) as a negative power of 3.

JavaScript code:

function f(n){

  var result = 0;

  for (var d=n; d<2*n; d++){

    var temp = 0;

    for (var NE=0; NE<2*n-d; NE++){

      temp += choose(n,NE);
    }

    result += choose(d - 1,d - n) * temp / Math.pow(3,d);
  }

  return 2 * result;
 }

function choose(n,k){
  if (k == 0 || n == k){
    return 1;
  }
  var product = n;
  for (var i=2; i<=k; i++){
    product *= (n + 1 - i) / i
  }
  return product;
}

Output:

for (var i=1; i<8; i++){
  console.log("F(" + i + "," + i + ") = " + f(i));
}

F(1,1) = 0.6666666666666666
F(2,2) = 0.8148148148148148
F(3,3) = 0.8641975308641975
F(4,4) = 0.8879743941472337
F(5,5) = 0.9024030889600163
F(6,6) = 0.9123609205913732
F(7,7) = 0.9197747256986194