Narcissistic half-and-half power-sequence terraces for Zn with n=pqt

AbstractA power-sequence terrace for Zn is a Zn terrace that can be partitioned into segments one of which contains merely the zero element of Zn whilst each other segment is either (a) a sequence of successive powers of an element of Zn, or (b) such a sequence multiplied throughout by a constant. If n is odd, a Zn terrace (a1,a2,…,an) is a narcissistic half-and-half terrace if ai−ai−1=an+2−i−an+1−i for i=2,3,…,(n+1)/2. Constructions are provided for narcissistic half-and-half power-sequence terraces for Zn with n=pqt where p and q are distinct odd primes and t is a positive integer. All the constructions are for terraces with as few segments as possible. Attention is restricted to constructions covering values of n with n=pqt and n<300; terraces are provided for all such values except n=189. Particularly elegant constructions are available for n=275

Some z_n-1 terraces from z_n power-sequences, n being an odd prime power

A terrace for Zm is a particular type of sequence formed from the m elements of Zm. For m odd, many procedures are available for constructing power-sequence terraces for Zm; each terrace of this sort may be partitioned into segments, of which one contains merely the zero element of Zm, whereas every other segment is either a sequence of successive powers of an element of Zm or such a sequence multiplied throughout by a constant. We now refine this idea to show that, for m=n−1, where n is an odd prime power, there are many ways in which power-sequences in Zn can be used to arrange the elements of Zn \ {0} in a sequence of distinct entries i, 1 &#8804; i &#8804; m, usually in two or more segments, which becomes a terrace for Zm when interpreted modulo m instead of modulo n. Our constructions provide terraces for Zn-1 for all prime powers n satisfying 0 &#60; n &#60; 300 except for n = 125, 127 and 257