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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

Table of n, a(n) for n=1..39.

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

Cf. A233248, A064414.

Sequence in context: A286070 A033863 A033908 * A229996 A255245 A210369

Adjacent sequences:  A233243 A233244 A233245 * A233247 A233248 A233249

KEYWORD

nonn

AUTHOR

Brandon Avila and Tanya Khovanova, Dec 06 2013

STATUS

approved

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Last modified March 30 06:59 EDT 2020. Contains 333119 sequences. (Running on oeis4.)