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Sub Distance Sequence(Free project!) A sub-distance sequence level-n Fn of a sequence F is defined as following: If F(k) is kth term of F, then Fn(k) is kth term of Fn created by: Fn(k) = F(n+k+1) - F(k) From here, one can expand the sub-distance sequence: Given a set Si = {Si-1 U ni}, S0={}. FSi is said to be a sub-distance sequence level ni of FSi-1 with FSi(k) = FSi-1(ni+k+1) - FSi-1(k) The first property is that: Fn1,n2(k)=Fn2,n1(k). One practicing way of determining sequence F from the equation Fk(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,...,ak+1. If pj are the solutions of the equation g(x) = j (j>0), then F will take the form as below: F(i) = , i<=k+1. F(i) = h(pi-k-1) + F(i-k-1), i>k+1. Sub-distance sequence holds similar properties as subsequence. It will be updated later on.
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