User:Trieu/Sub distance sequence

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A sub-distance sequence level-n F[n] of a sequence F is defined as following:

If F(k) is k^th term of F, then F[n](k) is k^th term of F[n] created by: F[n](k) = F(n+k+1) - F(k)

From here, one can expand the sub-distance sequence: Given a set S_i = {S_i-1 U n_i}, S₀={}.

F[S_i] is said to be a sub-distance sequence level n_i of F[S_i-1] with F[S_i](k) = F[S_i-1](n_i+k+1) - F[S_i-1](k)

The first property is that: F[n₁,n₂](k)=F[n₂,n₁](k).

One practicing way of determining sequence F from the equation F[k](g(x)) = h(x), where k is constant, g, h are functions of variable x is:

F will be contained k + 1 free parameters, say: a₁,a₂,...,a_k+1.

If p_j are the solutions of the equation g(x) = j (j>0), then F will take the form as below:

F(i) = $\sum _{i=1}^{k+1}a_{i}$ , i<=k+1.

F(i) = h(p_i-k-1) + F(i-k-1), i>k+1.

Sub-distance sequence holds similar properties as subsequence. It will be updated later on.