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%I #19 May 30 2023 19:43:53
%S 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 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 20,30,30,36,36,16,0,28,28,28
%N 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 See Dalfó and Fiol (2019) or A360000 for the definition of the 2-Fibonacci digraph.
%C 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 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 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 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 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 T(n,k) = A006206(k) for n >= k-1.
%e Triangle begins:
%e n\k| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
%e ---+-----------------------------------------------------
%e 1 | 1 1
%e 2 | 1 1 1
%e 3 | 1 1 1 1
%e 4 | 1 1 1 1 2 1 1
%e 5 | 1 1 1 1 2 2 1 1 2 2 2
%e 6 | 1 1 1 1 2 2 4 3 5 4 7 6 6 6 4 4 2 2
%o (Python)
%o import networkx as nx
%o from collections import Counter
%o def F(n): return nx.DiGraph(((0,0),(0,1),(1,0))) if n == 1 else nx.line_graph(F(n-1))
%o def A359997_row(n):
%o a = Counter(len(c) for c in nx.simple_cycles(F(n)))
%o return [a[k] for k in range(1,max(a)+1)]
%Y Cf. A006206 (main diagonal), A080023, A344018, A359998 (last element in each row), A359999, A360000 (row sums).
%K nonn,tabf
%O 1,14
%A _Pontus von Brömssen_, Jan 21 2023
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