How to calculate this sequence iteratively?

Question

I want calculate this sequence iteratively.

A(0,j)=j+1
A(i,0)=A(i-1,0)
A(i,j)=A(i-1,A(i,j-1))

This is my try

    public function calculsuite1Action($i,$j)
{
    $A = array();
    for ($k = 0; $k <= $i * $i * $i + $j; $k++) {
        $A[0][$k] = $k + 1;
    }
    for ($c = 1; $c <= $i; $c++) {
        for ($k = 0; $k <= $i * $i * $i + $j - $c; $k++) {
            if ($k == 0) {
                $A[$c][$k] = $A[$c - 1][1];
            } else {
                $A[$c][$k] = $A[$c - 1][$A[$c][$k - 1]];
            }
            if ($c == $i && $k == $j) {
                return $A[$i][$j];
            }

        }
    }
}

I'm trying to find a solution using PHP or any other programming language. How can I do that?

Answer 1

I have tried looking for a pattern for the first few entries. If I haven´t done any mistakes the sequence is pretty simple. It boils down to

A(i, j) = j+1

Just used this JavaScript program to verify I didn't make any mistakes

 for (i = 0; i < 5; i++){ for (j = 0; j < 5; j++){ console.log("A("+i+", "+j+") = "+calc(i,j)); } } function calc(i, j){ if(i==0) return j+1; else if(j == 0) return calc(i-1, 0); else return calc(i-1, calc(i, j-1)); } 

Answer 2

Here's a stack implementation. Please see here for an explanation of how I converted the recursion.

JavaScript code:

 /** * A(0,j)=j+1 * A(i,0)=A(i-1,0) * A(i,j)=A(i-1,A(i,j-1)) ***/ // Recursive function f(i,j){ if (i == 0) return j+1 else if (j == 0) return f(i-1,0) else return f(i-1,f(i,j-1)) } // Iterative function iterative(_i,_j){ let result = undefined; let stack = [[f, [_i, _j]]]; function g(i){ let Aij_1 = result; stack.push([f, [i-1, Aij_1]]); } function f(i, j){ if (!i){ result = j + 1; } else if (!j){ stack.push([f, [i-1, 0]]); } else { stack.push([g, [i]]); stack.push([f, [i, j-1]]); } } while (stack.length){ [func, params] = stack.pop(); func.apply(func, params); } return result; } for (let i=0; i<11; i++){ for (let j=0; j<11; j++) // Log both iterative and recursive results for comparison console.log( '(' + i + ', ' + j + ') => ' + JSON.stringify([iterative(i,j), f(i,j)]) ); }