OFFSET

1,14

COMMENTS

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.

LINKS

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.

FORMULA

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

EXAMPLE

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

PROG

(Python)

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)]

CROSSREFS

KEYWORD

nonn,tabf

AUTHOR

Pontus von Brömssen, Jan 21 2023

STATUS

approved