A360000 Number of directed cycles in the 2-Fibonacci digraph of order n.

2, 3, 4, 8, 16, 61, 437, 17766, 5885824, 111327315589

Offset: 1

Author

Pontus von Brömssen, Jan 21 2023

Keywords

• nonn
• more

Comments

The 2-Fibonacci digraph of order n, F(n), is defined by Dalfó and Fiol (2019). It can be defined as an iterated line digraph, where F(1) has two nodes, one directed edge in each direction between them, and a loop at one of the nodes, and for n >= 2 F(n) is the line digraph of F(n-1). (Compare the related de Bruijn graph, where the graph of order one has loops at both nodes.) Its nodes can be identified with binary sequences of length n with no adjacent 1's (or fibbinary numbers below 2^n if the nodes are labeled by integers instead of binary sequences), with a directed edge from (x_0, ..., x_{n-1}) to (x_1, ..., x_n) if there are no consecutive 1's in (x_0, ..., x_n). For n >= 2, it is also the subgraph of the de Bruijn graph (of the same order) induced by the nodes with no adjacent 1's. It has A000045(n+2) nodes and A000045(n+3) edges.

Equivalently, a(n) is the number of cycles with no adjacent 1's that can be produced by a general n-stage feedback shift register.

Examples

			For n = 4 there are a(4) = 8 cycles:
  0000 -> 0000;
  0101 -> 1010 -> 0101;
  0010 -> 0100 -> 1001 -> 0010;
  0001 -> 0010 -> 0100 -> 1000 -> 0001;
  0000 -> 0001 -> 0010 -> 0100 -> 1000 -> 0000;
  0010 -> 0101 -> 1010 -> 0100 -> 1001 -> 0010;
  0001 -> 0010 -> 0101 -> 1010 -> 0100 -> 1000 -> 0001;
  0000 -> 0001 -> 0010 -> 0101 -> 1010 -> 0100 -> 1000 -> 0000.
		

Links

• C. Dalfó and M. A. Fiol, On d-Fibonacci digraphs, arXiv:1909.06766 [math.CO], 2019.

Crossrefs

Cf. A000045, A306522, A359997, A359998, A359999.

Programs

Python

import networkx as nx
def F(n): return nx.DiGraph(((0,0),(0,1),(1,0))) if n == 1 else nx.line_graph(F(n-1))
def A360000(n): return sum(1 for c in nx.simple_cycles(F(n)))

Extensions

a(10) from Bert Dobbelaere, Jan 24 2023