%I A137725
%S A137725 4,4,16,18,24,32,46,58,82,112,158,220,316,450,650,938,1364,1982,2892,4220,
               6170,9022,13206,
%T A137725 19332,28314,41472,60760,89022,130446,191150,280120,410506,601600,881656,
               1292102,
%U A137725 1893634,2775226,4067256,5960822,8735972,12803156,18763898,27499794,40302866,
               59066684
%N A137725 Number of sequences of length n with elements {-2,-1,+1,+2}, such that 
               the sum of elements of the whole sequence but of no proper subsequence 
               equals 0 modulo n. For n>=4, the number of Hamiltonian (directed) 
               circuits on the circulant graph C_n(1,2).
%C A137725 Number of 1-D walks with jumps to next-nearest neighbors with n steps, 
               starting at 0 and ending at -2n, -n, 0, n, or 2n, such that every 
               point is visited at most once and every pair of points at the distance 
               n contains at least one unvisited point (not counting the ending 
               visit). Cf. A092765.
%C A137725 For n>1, the number of circular permutations (counted up to rotations) 
               of {0, 1,...,n-1} such that the distance between every two adjacent 
               elements is -2,-1,1,or 2 modulo n. Cf. A003274.
%H A137725 Mordecai J. Golin and Yiu Cho Leung, <a href="http://www.cse.ust.hk/tcsc/
               RR/2004-02.ps.gz">Unhooking Circulant Graphs: A Combinatorial Method 
               for Counting Spanning Trees, Hamiltonian Cycles and other Parameters</
               a>. Technical report HKUST-TCSC-2004-02.
%H A137725 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CirculantGraph.html">Circulant Graph</a>
%H A137725 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HamiltonianCircuit.html">Hamiltonian Circuit</a>.
%F A137725 For even n>=4, a(n) = 2*(n + 3*A000930(n) - 2*A000930(n)); for odd n>
               =3, a(n) = 2*(n + 3*A000930(n) - 2*A000930(n)+1).
%F A137725 For n>8, a(n) = 2*a(n-1) - a(n-3) - a(n-5) + a(n-6) or a(n) = a(n-1) 
               + a(n-2) - a(n-5) - 4.
%F A137725 O.g.f.: -2*x^2-2*x-6-1/(x+1)+2/(x-1)^2+1/(x-1)+(4*x-6)/(x^3+x-1). - R. 
               J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 10 2008
%Y A137725 Cf. A124353, A137726
%Y A137725 Sequence in context: A164111 A164906 A129884 this_sequence A082649 A156232 
               A053441
%Y A137725 Adjacent sequences: A137722 A137723 A137724 this_sequence A137726 A137727 
               A137728
%K A137725 nonn
%O A137725 1,1
%A A137725 Max Alekseyev (maxale(AT)gmail.com), Feb 8, 2008