A233246 -id:A233246 - cp's OEIS frontend

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

Brandon Avila and Tanya Khovanova, Dec 06 2013

nonn

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.

			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.

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

Cf. A233246, A064414

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}]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica