A233248 The average cycle length of cycles in Fibonacci-like sequences modulo n over all starting pairs of remainders.

1, 2, 7, 5, 17, 18, 16, 10, 22, 42, 10, 21, 28, 39, 32, 21, 36, 23, 18, 48, 16, 24, 48, 22, 84, 70, 66, 45, 14, 79, 30, 41, 36, 36, 66, 24, 76, 18, 53, 50, 40, 40, 88, 28, 93, 48, 32, 24, 110, 210, 68, 80, 108, 67, 20, 47, 66, 34, 58, 91, 60, 30

Offset

1,2

Comments

a(n) = Round(A233246(n)/n^2.

If n is in A064414, then a(n) is the average distance between two neighboring multiples of n.

If n is in A064414, then a(n)/2 is the average distance to the next zero over all starting pairs of remainders.

Links

Table of n, a(n) for n=1..62.

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)=Round((1+9+36+36)/16)=5.

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[Round[
  Total[Flatten[Table[cl[i, j, n], {i, 0, n - 1}, {j, 0, n - 1}]]]/
   n^2], {n, 70}]

Crossrefs

Cf. A233246, A064414

Sequence in context: A096037 A286379 A343314 * A114025 A135566 A100759

Adjacent sequences: A233245 A233246 A233247 * A233249 A233250 A233251

Keyword

nonn

Author

Brandon Avila and Tanya Khovanova, Dec 06 2013

Status

approved