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A233246 Sum of squares of cycle lengths for different cycles in Fibonacci-like sequences modulo n. 1
1, 10, 65, 82, 417, 650, 769, 658, 1793, 4170, 1151, 3026, 4705, 7690, 7137, 5266, 10369, 7562, 6319, 19218, 6977, 11510, 25345, 12818, 52417, 47050, 48449, 35410, 11565, 71370, 28351, 42130, 39615, 41482, 81057, 30674, 103969, 25282, 80033 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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).
This sequence divided by n^2 is the average cycle length per different starting pairs modulo n, see A233248.
If n is in A064414, then a(n)/n^2 is the average distance between two neighboring multiples of n.
If n is in A064414, then a(n)/2n^2 is the average distance to the next zero over all starting pairs of remainders.
LINKS
B. Avila and T. Khovanova, Free Fibonacci Sequences, arXiv preprint arXiv:1403.4614, 2014 and J. Int. Seq. 17 (2014) # 14.8.5
EXAMPLE
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.
MATHEMATICA
cl[i_, j_, n_] := (step = 1; first = i; second = j;
next = Mod[first + second, n];
While[second != i || next != j, step++; first = second;
second = next; next = Mod[first + second, n]]; step)
Table[Total[
Flatten[Table[cl[i, j, n], {i, 0, n - 1}, {j, 0, n - 1}]]], {n, 50}]
CROSSREFS
Sequence in context: A286070 A033908 A033863 * A229996 A255245 A210369
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified June 17 14:56 EDT 2024. Contains 373448 sequences. (Running on oeis4.)