Camel sequences and quadratic residues

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 distribute obtained n2 numbers between 2n "boxes" 1; 2; :::; 2n 1; 2n = 0 (mod 2n). The average cardinality of a box is n2 2n = n2 . 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 (bn2cn 1 (n 1) bn2cn 1 n) and (dn2 en 1 1 dn2 en 1 0) 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 rst sequence above is down-camel, and the second one is up-camel. Camel sequences have applications in extremal graph theory. Here we prove that there are in nitely 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. We conjecture that there are no other camel sequences.