A359997 - OEIS

%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