A213277 a(n) is the length of the cycle of first differences of k such that Fibonacci(k) mod n = k mod n.

3, 8, 3, 8, 6, 16, 3, 8, 14, 10, 4, 28, 24, 16, 3, 36, 6, 18, 11, 16, 15, 48, 3, 18, 42, 8, 12, 14, 14, 30, 3, 40, 18, 32, 4, 76, 9, 56, 11, 40, 12, 88, 15, 16, 24, 32, 3, 16, 34, 24, 21, 108, 6, 8, 6, 24, 21, 58, 12, 60, 15, 16, 3, 56, 30, 136, 9, 16, 56

Offset: 2

Author

Lars Blomberg, Jun 08 2012

Keywords

• nonn

Comments

In calculating the terms, a set of values A that is found twice (AA) is not enough to be certain that A is a cycle since the continuation may be AAA..AAABAAA..AAAB where B is a different set of values. In calculating the data above, a cycle is accepted when it has occurred 10 times in a row.

Examples

			Example with n=3:
Fib(k): 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, ...
Fib(k) mod 3: 0,1,1,2,0,2,2,1,0,1,1,2,0,2,2,1,0,1,1,2,0,2,2,1,0
k mod 3:      0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0,1,2,0
Accepted k values indicated by x:
              x,x,-,-,-,x,-,x,-,-,x,x,x,-,-,-,-,-,-,-,-,-,-,-,x
Accepted k values: 0, 1, 5, 7, 10, 11, 12, 14, 24
First differences of k values: 1, 4, 2, 3, 1, 1, 2, 10
After this the cycle repeats, so a(3) = 8.
		

Links

• Lars Blomberg, Table of n, a(n) for n = 2..23002

Crossrefs

Cf. A000045, A213278.