OFFSET
1,1
COMMENTS
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.
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.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Mordecai J. Golin and Yiu Cho Leung, Unhooking Circulant Graphs: A Combinatorial Method for Counting Spanning Trees, Hamiltonian Cycles and other Parameters. Technical report HKUST-TCSC-2004-02.
Eric Weisstein's World of Mathematics, Circulant Graph
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,0,-1,1).
FORMULA
For even n>=4, a(n) = 2*(n + 3*A000930(n) - 2*A000930(n-1)); for odd n>=3, a(n) = 2*(n + 1 + 3*A000930(n) - 2*A000930(n-1)).
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.
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, Feb 10 2008
MATHEMATICA
CoefficientList[Series[-2*x^2-2*x-6-1/(x+1)+2/(x-1)^2+1/(x-1)+(4*x-6)/(x^3+x-1), {x, 0, 50}], x] (* G. C. Greubel, apr 28 2017 *)
PROG
(PARI) x='x+O('x^50); Vec(-2*x^2-2*x-6-1/(x+1)+2/(x-1)^2+1/(x-1)+(4*x-6)/(x^3+x-1)) \\ G. C. Greubel, Apr 28 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Max Alekseyev, Feb 08 2008
EXTENSIONS
Typo in formulas corrected by Max Alekseyev, Nov 03 2010
STATUS
approved