stairs.cs

void Main()
{
	Steps(0,5).Dump();
}

Dictionary<int,int> memo = new Dictionary<int,int>();

int Steps(int step, int numstairs) 
{
	if (numstairs == 0)
		return 1;
		
	if (numstairs < 0)
		return 0;

	if (memo.TryGetValue(numstairs, out int val))
		return val;

	var steps = Steps(1, numstairs - 1) +
		   Steps(2, numstairs - 2) +
           Steps(3, numstairs - 3);

	memo.Add(numstairs, steps);
	//Console.WriteLine($"{numstairs}<={steps}");
	return steps;
}
// Stairs(0,5) returns 13