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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.

%I A359997 #19 May 30 2023 19:43:53
%S A359997 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,
%T A359997 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,
%U A359997 20,30,30,36,36,16,0,28,28,28
%N 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.
%C A359997 See Dalfó and Fiol (2019) or A360000 for the definition of the 2-Fibonacci digraph.
%C A359997 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.
%C A359997 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).
%C A359997 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.
%H A359997 Pontus von Brömssen, <a href="/A359997/b359997.txt">Table of n, a(n) for n = 1..320</a> (rows n = 1..10; row 10 computed by Bert Dobbelaere)
%H A359997 C. Dalfó and M. A. Fiol, <a href="https://arxiv.org/abs/1909.06766">On d-Fibonacci digraphs</a>, arXiv:1909.06766 [math.CO], 2019.
%F A359997 T(n,k) = A006206(k) for n >= k-1.
%e A359997 Triangle begins:
%e A359997   n\k| 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18
%e A359997   ---+-----------------------------------------------------
%e A359997   1  | 1  1
%e A359997   2  | 1  1  1
%e A359997   3  | 1  1  1  1
%e A359997   4  | 1  1  1  1  2  1  1
%e A359997   5  | 1  1  1  1  2  2  1  1  2  2  2
%e A359997   6  | 1  1  1  1  2  2  4  3  5  4  7  6  6  6  4  4  2  2
%o A359997 (Python)
%o A359997 import networkx as nx
%o A359997 from collections import Counter
%o A359997 def F(n): return nx.DiGraph(((0,0),(0,1),(1,0))) if n == 1 else nx.line_graph(F(n-1))
%o A359997 def A359997_row(n):
%o A359997     a = Counter(len(c) for c in nx.simple_cycles(F(n)))
%o A359997     return [a[k] for k in range(1,max(a)+1)]
%Y A359997 Cf. A006206 (main diagonal), A080023, A344018, A359998 (last element in each row), A359999, A360000 (row sums).
%K A359997 nonn,tabf
%O A359997 1,14
%A A359997 _Pontus von Brömssen_, Jan 21 2023