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%I A233248 #16 Feb 28 2018 15:19:59
%S A233248 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 A233248 84,70,66,45,14,79,30,41,36,36,66,24,76,18,53,50,40,40,88,28,93,48,32,
%U A233248 24,110,210,68,80,108,67,20,47,66,34,58,91,60,30
%N A233248 The average cycle length of cycles in Fibonacci-like sequences modulo n over all starting pairs of remainders.
%C A233248 a(n) = Round(A233246(n)/n^2.
%C A233248 If n is in A064414, then a(n) is the average distance between two neighboring multiples of n.
%C A233248 If n is in A064414, then a(n)/2 is the average distance to the next zero over all starting pairs of remainders.
%H A233248 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 A233248 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 A233248 cl[i_, j_, n_] := (step = 1; first = i; second = j;
%t A233248   next = Mod[first + second, n];
%t A233248   While[second != i || next != j, step++; first = second;
%t A233248    second = next; next = Mod[first + second, n]]; step)
%t A233248 Table[Round[
%t A233248   Total[Flatten[Table[cl[i, j, n], {i, 0, n - 1}, {j, 0, n - 1}]]]/
%t A233248    n^2], {n, 70}]
%Y A233248 Cf. A233246, A064414
%K A233248 nonn
%O A233248 1,2
%A A233248 _Brandon Avila_ and _Tanya Khovanova_, Dec 06 2013

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