Camel sequences and quadratic residues Original Research Article

Given an even cyclic (+1,−1) sequence s=(s0,…,s2n−1) which consists of n plus ones and n minus ones, let us compute i+j (mod 2n) for all n2 (+1,−1)-pairs (si,sj) and insert the obtained n2 numbers into 2n “boxes” b0,b1,…,b2n−1, where box bk contains the multiplicity of k. The cyclic sequence B(s)=(b0,b1,…,b2n−1) is referred to as the box distribution of s. The average cardinality of a box is n2/2n=n/2. Some sequences have quite remarkable box-distributions, “almost average everywhere with two big humps”. For example, n=5, s=(+1+1−1−1+1−1+1−1−1+1), B=(3333133330)=(341 340); n=7, s=(+1+1+1−1+1−1−1−1+1+1−1+1−1−1), B=(33333363333337)=(366 367). In general, given an odd n, the box-distributions (⌊n/2⌋n−1(n−1)⌊n/2⌋n−1n) and (⌈n/2⌉n−11⌈n/2⌉n−10) as well as the sequences which generate them, will be called the camel distributions and camel sequences, respectively, up-camel and down-camel. For example, the first sequence above is down-camel, and the second one is up-camel. Here we prove that there are infinitely many ‘camels’ of both types. More precisely, for every prime n=4j−1 we construct an up-camel sequence and for every prime n=4j+1 a down-camel one. In both cases these sequences are related to quadratic residues and non-residues modulo n. Camel sequences have applications in extremal graph theory.