# Creating Recursive Sequences

Quite a simple question, I reckon, however, even quite an extensive search hasn't helped me.

I want to define a recursively defined sequence that starts with defined f[1] and f[2] and distinguishes the input like so: f[3k], f[3k+1], f[3k+2]. Each of these would have its own expression.

For further clarification, an example: $$f(3n) = f(n)+1$$ $$f(3n+1) = f(n)+2$$ $$f(3n+2) = f(n)+3$$ How does one go about that? I do know how to write simple recursion formulas on the level of Fibonacci or a factorial, it's only the distinguishing the input part. Also, this should be possible without any if statements, etc.

Thanks for any help!

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You can simply define an expression for each such pattern, using Condition (shorthand: /;) to specify that it should only match in the appropriate cases. I'm assuming by f[3k] you mean some input that's fully divisible by 3, so I'd write:

f[1] = 0;
f[2] = 0;
f[x_] /; Mod[x, 3] === 0 := {"3k def",f[x-1]}
f[x_] /; Mod[x - 1, 3] === 0 := {"3k def",f[x-1]}
f[x_] /; Mod[x - 2, 3] === 0 := "{"3k def",f[x-1]}

f[6]
(* {"3k def", {"3k+2 def", {"3k+1 def", {"3k def", 0}}}} *)

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Sorry I wasn't clear enough, however, this looks like it should be the solution! Thanks, I shall try that now and then report. –  Dahn Jahn Jan 28 '13 at 10:56
Yes, this is exactly what I needed. Thanks! –  Dahn Jahn Jan 28 '13 at 11:19