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

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

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 A033908 A033863 * A229996 A255245 A210369

Adjacent sequences: A233243 A233244 A233245 * A233247 A233248 A233249

Keyword

nonn

Author

Brandon Avila and Tanya Khovanova, Dec 06 2013

Status

approved