Some da capo directed power-sequence Zn+1Zn+1 terraces with n an odd prime power

A terrace for ZnZn is a particular type of sequence formed from the n   elements of ZnZn. For n   odd, many procedures are available for constructing power-sequence terraces for ZnZn; each such terrace may be partitioned into segments one of which contains merely the zero element of ZnZn, whereas each other segment is either (a) a sequence of successive powers of a non-zero element of ZnZn or (b) such a sequence multiplied throughout by a constant. We now extend this idea by using power-sequences in ZnZn to produce some terraces for Zn+1Zn+1 where n   is an odd prime power satisfying n≡1n≡1 or 3 (mod 8). Each terrace now consists of a sequence of segments, one containing merely the element 0 and another merely containing the element n, the remaining segments each being of type (a) or (b) above with each of its distinct entries i   from Zn⧹{0}Zn⧹{0} evaluated so that 1⩽i⩽n-11⩽i⩽n-1. The terraces constructed are da capo   directed terraces, i.e. each terrace (a1,a2,…,an+1)(a1,a2,…,an+1) has ai-ai-1=-(ai+m-ai-1+mai-ai-1=-(ai+m-ai-1+m) for all i   satisfying 2≤i≤m2≤i≤m where m=(n+1)/2m=(n+1)/2.