It's easy to show that the conjectured formula satisfies the requirements. Did you prove that it is unique? I'd like to see what you did.
Of course! I'm a beginner at proofs, I tried to prove it with strong induction.
Conjecture:
Closed form of reccurence sequence [MATH]x_{m+n}+x_{m-n}=1/2(x_{2m}+x_{2n})[/MATH] for m,n∈ N+{0}, m>=n
is: [MATH]x_{k}=k^2[/MATH] for k>=0
Proof:
1. Base case: k=0
Let m=0, n=0, then [MATH]x_{0}=0[/MATH] is the only solution to [MATH]2x_0=x_0[/MATH][MATH]0^2=0[/MATH], holds.
2. By strong induction, assume that formula holds for [MATH]1,2,3..,k-1,k[/MATH], then it has to hold for [MATH]k+1[/MATH].
Let m=k, n=1
[MATH]x_{k+1}=1/2(x_{2k}+x_2)-x_{k-1}[/MATH]Let m=f,n=0
[MATH]x_{2k}=4x_k[/MATH]
Substituting that, [MATH]x_2=4[/MATH], [MATH]x_k=k^2[/MATH] and [MATH]x_{k-1}=(k-1)^2[/MATH],
we get [MATH]x_{k+1}=(k+1)^2[/MATH].
Proven