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A233246 Sum of squares of cycle lengths for different cycles in Fibonacci-like sequences modulo n.

%I A233246 #15 Feb 28 2018 15:20:20
%S A233246 1,10,65,82,417,650,769,658,1793,4170,1151,3026,4705,7690,7137,5266,
%T A233246 10369,7562,6319,19218,6977,11510,25345,12818,52417,47050,48449,35410,
%U A233246 11565,71370,28351,42130,39615,41482,81057,30674,103969,25282,80033
%N A233246 Sum of squares of cycle lengths for different cycles in Fibonacci-like sequences modulo n.
%C A233246 Here Fibonacci-like means a sequence following the Fibonacci recursion: b(n)=b(n-1)+b(n-2). These sequences modulo n cycle. The number of different cycles is A015134(n).
%C A233246 This sequence divided by n^2 is the average cycle length per different starting pairs modulo n, see A233248.
%C A233246 If n is in A064414, then a(n)/n^2 is the average distance between two neighboring multiples of n.
%C A233246 If n is in A064414, then a(n)/2n^2 is the average distance to the next zero over all starting pairs of remainders.
%H A233246 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 A233246 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)=1+9+36+36=82.
%t A233246 cl[i_, j_, n_] := (step = 1; first = i; second = j;
%t A233246   next = Mod[first + second, n];
%t A233246   While[second != i || next != j, step++; first = second;
%t A233246    second = next; next = Mod[first + second, n]]; step)
%t A233246 Table[Total[
%t A233246   Flatten[Table[cl[i, j, n], {i, 0, n - 1}, {j, 0, n - 1}]]], {n, 50}]
%Y A233246 Cf. A233248, A064414.
%K A233246 nonn
%O A233246 1,2
%A A233246 _Brandon Avila_ and _Tanya Khovanova_, Dec 06 2013