After trying to understand your description of how it should be the recursion, i have come to believe that what you need is something like the following

k[0] := 0;

k[1] := 1;

k[2] := 2;

k[3] := 3;

k[n_] := 1 + k[n - 1] /; n > 3;

I tried the following values to try to make sure that works well the recursion,

Table[k[i], {i, {4, 5, 9, 13, 7, 11, 13, 17, 43, 47, 53, 59, 61, 67, 
   71}}]

I think that the following link may help you to understand more things about how to solve your problem, applied the same idea to your project.

Mathematica, Fibonacci sequence using While and Module commands

Hi Frida, I'd like to be able to help you to solve your problem, but not for what is the algorithm that you portray, it would be useful something of prior information, i think you are missing some points and commas at the end of the assignments, you should do something like this.:

K[1] := 1 + K[1-1] ;

folglich =  1 + K[0];

K[2] := 1 + K[2-1] ;

K[3] := 1 + K[3-1];

taking advantage of the opportunity I would like to ask you where you define k[0], since you call it and not defined, I hope your answer.

Greetings