A344018 -id:A344018 - cp's OEIS frontend

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.

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

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Formula