0,2

Define a graph with 2n vertices. Vertices 1 through n will be on the top half, vertices n+1 through 2n will be on the bottom half. For 1 <= i < j <=n, create a directed edge from vertex i to vertex j if j-i is either 1 or 2. For n+1<=i<j<=2n, create a directed edge from vertex j to vertex i if j-i is either 1 or 2. Lastly, create a directed edge from i to n+i and vice versa for 1 <= i <= n. (A graph of this general type is called a hamburger.) The value a(n) gives the number of vertex-disjoint unions of directed cycles in this graph. Also calculable as the determinant of an n X n Toeplitz matrix.

For n>=0, the set {A112832(2n+1)} is the intersection of the set of numbers of the form a^2+(a+1)^2 and the set of numbers of the form b^2+5(b+1)^2. - Kieren MacMillan (kieren(AT)alumni.rice.edu), Dec 19 2007

C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.

G.f.: A(x) = (1-2*x-3*x^2+x^3)/(1-4*x+4*x^3-x^4)

The number of non-intersecting cycle systems in the particular directed graph of order 4 is 61.

f:=n->transpose(ToeplitzMatrix([seq(0, i=1..n-3), -1, -1, 2, seq(fibonacci(i), i=2..n)])); A:=[1, 2, 5, seq(det(f(i)), i=3..25)];

See also A112831.

Sequence in context: A150009 A150010 A150011 this_sequence A148415 A148416 A003456

Adjacent sequences: A112829 A112830 A112831 this_sequence A112833 A112834 A112835

easy,nonn

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005