A359999 -id:A359999 - cp's OEIS frontend

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

Pontus von Brömssen, Jan 21 2023

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.

			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.

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

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

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)))

a(10) from Bert Dobbelaere, Jan 24 2023

A359997 Irregular triangle read by rows: T(n,k) is the number of directed cycles of length k in the 2-Fibonacci digraph of order n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 4, 3, 5, 4, 7, 6, 6, 6, 4, 4, 2, 2, 1, 1, 1, 1, 2, 2, 4, 5, 5, 6, 8, 10, 15, 20, 20, 24, 23, 19, 18, 20, 30, 30, 36, 36, 16, 0, 28, 28, 28

Offset: 1

Pontus von Brömssen, Jan 21 2023

See Dalfó and Fiol (2019) or A360000 for the definition of the 2-Fibonacci digraph.
Equivalently, T(n,k) is the number of cycles of length k with no adjacent 1's that can be produced by a general n-stage feedback shift register.
Apparently, the number of terms in the n-th row (i.e., the length of the longest cycle in the 2-Fibonacci digraph of order n) is A080023(n).
Interestingly, the 2-Fibonacci digraph of order 7 has cycles of all lengths from 1 up to the maximum 29, except 26. For all other orders n <= 10, there are no such gaps, i.e., the graph is weakly pancyclic.

			Triangle begins:
  n\k| 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18
  ---+-----------------------------------------------------
  1  | 1  1
  2  | 1  1  1
  3  | 1  1  1  1
  4  | 1  1  1  1  2  1  1
  5  | 1  1  1  1  2  2  1  1  2  2  2
  6  | 1  1  1  1  2  2  4  3  5  4  7  6  6  6  4  4  2  2

Pontus von Brömssen, Table of n, a(n) for n = 1..320 (rows n = 1..10; row 10 computed by Bert Dobbelaere)
C. Dalfó and M. A. Fiol, On d-Fibonacci digraphs, arXiv:1909.06766 [math.CO], 2019.

Cf. A006206 (main diagonal), A080023, A344018, A359998 (last element in each row), A359999, A360000 (row sums).

import networkx as nx
from collections import Counter
def F(n): return nx.DiGraph(((0,0),(0,1),(1,0))) if n == 1 else nx.line_graph(F(n-1))
def A359997_row(n):
    a = Counter(len(c) for c in nx.simple_cycles(F(n)))
    return [a[k] for k in range(1,max(a)+1)]

T(n,k) = A006206(k) for n >= k-1.

cp's OEIS Frontend

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

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