A360000 -id:A360000 - 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.

A359994 Independence number of the 2-Fibonacci digraph of order n.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 16, 25, 44, 67, 115

Offset: 1

Pontus von Brömssen, Jan 21 2023

See Dalfó and Fiol (2019) or A360000 for the definition of the 2-Fibonacci graph.

a(12) >= 179. - Pontus von Brömssen, Nov 12 2023

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

Cf. A006946, A359995, A359996, A360000.

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 A359994(n): return nx.max_weight_clique(nx.complement(nx.Graph(F(n))),weight=None)[1]

A359995 Number of maximal independent sets in the 2-Fibonacci digraph of order n.

Original entry on oeis.org

2, 3, 4, 10, 32, 184, 5110, 681454

Offset: 1

Pontus von Brömssen, Jan 21 2023

See Dalfó and Fiol (2019) or A360000 for the definition of the 2-Fibonacci graph.

The loop at node 0 is disregarded, so 0 is allowed in the independent sets.

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

Cf. A333077, A359994, A359996, A360000.

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 A359995(n): return sum(1 for c in nx.find_cliques(nx.complement(nx.Graph(F(n)))))

A359996 Number of maximum independent sets in the 2-Fibonacci digraph of order n.

Original entry on oeis.org

2, 3, 4, 10, 3, 42, 12, 706

Offset: 1

Pontus von Brömssen, Jan 21 2023

See Dalfó and Fiol (2019) or A360000 for the definition of the 2-Fibonacci graph.

The loop at node 0 is disregarded, so 0 is allowed in the independent sets.

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

Cf. A333078, A359994, A359995, A360000.

A359998 Number of longest directed cycles in the 2-Fibonacci digraph of order n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 28, 216, 65200, 167084480

Offset: 1

Pontus von Brömssen, Jan 21 2023

See Dalfó and Fiol (2019) or A360000 for the definition of the 2-Fibonacci digraph.

The longest cycles appear to have length A080023(n).

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

Cf. A016031, A080023, A359997, A360000.

a(10) from Bert Dobbelaere, Jan 24 2023

A359999 Most common cycle length in the 2-Fibonacci digraph of order n. In case of ties, choose the largest length.

Original entry on oeis.org

2, 3, 4, 5, 11, 11, 24, 32, 59, 92

Offset: 1

Pontus von Brömssen, Jan 21 2023

See Dalfó and Fiol (2019) or A360000 for the definition of the 2-Fibonacci graph.

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

Cf. A359997, A360000.

a(10) from Bert Dobbelaere, Jan 24 2023

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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.

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A359994 Independence number of the 2-Fibonacci digraph of order n.

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A359995 Number of maximal independent sets in the 2-Fibonacci digraph of order n.

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A359996 Number of maximum independent sets in the 2-Fibonacci digraph of order n.

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A359998 Number of longest directed cycles in the 2-Fibonacci digraph of order n.

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A359999 Most common cycle length in the 2-Fibonacci digraph of order n. In case of ties, choose the largest length.

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