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A233248 The average cycle length of cycles in Fibonacci-like sequences modulo n over all starting pairs of remainders. 2

%I

%S 1,2,7,5,17,18,16,10,22,42,10,21,28,39,32,21,36,23,18,48,16,24,48,22,

%T 84,70,66,45,14,79,30,41,36,36,66,24,76,18,53,50,40,40,88,28,93,48,32,

%U 24,110,210,68,80,108,67,20,47,66,34,58,91,60,30

%N The average cycle length of cycles in Fibonacci-like sequences modulo n over all starting pairs of remainders.

%C a(n) = Round(A233246(n)/n^2.

%C If n is in A064414, then a(n) is the average distance between two neighboring multiples of n.

%C If n is in A064414, then a(n)/2 is the average distance to the next zero over all starting pairs of remainders.

%H B. Avila and T. Khovanova, <a href="http://arxiv.org/abs/1403.4614">Free Fibonacci Sequences</a>, arXiv preprint arXiv:1403.4614, 2014 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Avila/avila4.html">J. Int. Seq. 17 (2014) # 14.8.5</a>

%e For n=4 there are four possible cycles: A trivial cycle of length 1: 0; two cycles of length 6: 0,1,1,2,3,1; and a cycle of length 3: 0,2,2. Hence, a(4)=Round((1+9+36+36)/16)=5.

%t cl[i_, j_, n_] := (step = 1; first = i; second = j;

%t next = Mod[first + second, n];

%t While[second != i || next != j, step++; first = second;

%t second = next; next = Mod[first + second, n]]; step)

%t Table[Round[

%t Total[Flatten[Table[cl[i, j, n], {i, 0, n - 1}, {j, 0, n - 1}]]]/

%t n^2], {n, 70}]

%Y Cf. A233246, A064414

%K nonn

%O 1,2

%A _Brandon Avila_ and _Tanya Khovanova_, Dec 06 2013

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Last modified May 28 21:37 EDT 2020. Contains 334690 sequences. (Running on oeis4.)