We would like to define $k$ in advance as an even number

Question

Can we define in advance that $k$ is even?

For example, when defining the sequence $\{a_n\}$ inductively as follows,

input:

a[n_]:=If[Mod[Abs[a[n-1]-a[n-2]],2]==0,a[n-1]-a[n-2],a[n-1]-2];
   a[1] = 3; a[2] = k; 

We would like to define $k$ in advance as an even number.

Answer 1

This is probably most easily accomplished as follows

$Assumptions = {khalf \[Element] Integers}
a[n_] := a[n] = FullSimplify[
    If[Mod[Abs[a[n - 1] - a[n - 2]], 2] == 0, 
       a[n - 1] - a[n - 2], 
       a[n - 1] - 2]];
a[1] = 3;
a[2] = 2 khalf

Note that

• I memoized a[n] to save on time.
• I wrapped the evaluation in FullSimplify to trigger the application of the $Assumptions.
• I changed a[2] to twice khalf and put into the $Assumptions that khalf is an integer.

Then

a[3] == 2(-1 + khalf)
a[4] == -2
a[5] == -2 khalf
a[6] == 2 - 2 khalf

and so on.