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

Author

Brandon Avila and Tanya Khovanova, Dec 06 2013

Keywords

• nonn

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.

Examples

			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.
		

Links

• B. Avila and T. Khovanova, Free Fibonacci Sequences, arXiv preprint arXiv:1403.4614, 2014 and J. Int. Seq. 17 (2014) # 14.8.5

Crossrefs

Cf. A233248, A064414.

Programs

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}]